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All Textbook Solutions for A First Course in Differential Equations with Modeling Applications (MindTap Course List)

In Problems 18 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). 1. (1 x)y 4xy + 5y = cos x In Problems 18 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). 2. xd3ydx3(dydx)4+y=0 In Problems 18 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). 3. t5y(4) t3y + 6y = 0 In Problems 18 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). 4. d2udr2+dudr+u=cos(r+u)In Problems 18 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). 5. d2ydx2=1+(dydx)2 In Problems 18 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). 6. d2Rdt2=kR2 In Problems 18 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). 7. (sin)y(cos)y=2 In Problems 18 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). 8. x(1x23)x+x=0 In Problems 9 and 10 determine whether the given first-order differential equation is linear in the indicated dependent variable by matching it with the first differential equation given in (7). 9. (y21)dx+xdy=0; in y; in x In Problems 9 and 10 determine whether the given first-order differential equation is linear in the indicated dependent variable by matching it with the first differential equation given in (7). 10. u dv + (v + uv ueu) du = 0; in v; in u In Problems 1114 verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution. 11. 2y + y = 0; y = ex/2 In Problems 1114 verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution. 12. dydt+20y=24;y=6565e20t In Problems 1114 verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution. 13. y 6y + 13y = 0; y = e3x cos 2x In Problems 1114 verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution. 14. y + y = tan x; y = (cos x) ln(sec x + tan x)In Problems 1518 verify that the indicated function y = (x) is an explicit solution of the given first-order differential equation. Proceed as in Example 6, by considering simply as a function and give its domain. Then by considering as a solution of the differential equation, give at least one interval I of definition. 15. (yx)y=yx+8;y=x+4x+2 In Problems 1518 verify that the indicated function y = (x) is an explicit solution of the given first-order differential equation. Proceed as in Example 6, by considering simply as a function and give its domain. Then by considering as a solution of the differential equation, give at least one interval I of definition. 16. y = 25 + y2; y = 5 tan 5x In Problems 1518 verify that the indicated function y = (x) is an explicit solution of the given first-order differential equation. Proceed as in Example 6, by considering simply as a function and give its domain. Then by considering as a solution of the differential equation, give at least one interval I of definition. 17. y = 2xy2; y = 1/(4 x2)In Problems 1518 verify that the indicated function y = (x) is an explicit solution of the given first-order differential equation. Proceed as in Example 6, by considering simply as a function and give its domain. Then by considering as a solution of the differential equation, give at least one interval I of definition. 18. 2y = y3 cos x; y = (1 sin x)1/2 In Problems 19 and 20 verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution y = (x) in each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval I of definition of each solution . 19. dXdt=(X1)(12X);ln(2X1X1)=t In Problems 19 and 20 verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution y = (x) in each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval I of definition of each solution . 20. 2xy dx + (x2 y)dy = 0; 2x2y + y2 = 1 In Problems 2124 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution. 21. dPdt=P(1P);P=c1et1+c1et In Problems 2124 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution. 22. dydx+4xy=8x3;y=2x21+c1e2x2 In Problems 2124 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution. 23. d2ydx24dydx+4y=0;y=c1e2x+c2xe2x In Problems 2124 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution. 24. x3d3ydx3+2x2d2ydx2xdydx+y=12x2;y=c1x1+c2x+c3xlnx+4x2 In Problems 2528 use (12) to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution. 25. xdydx3xy=1;y=e3x1xe3ttdt In Problems 2528 use (12) to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution. 26. 2xdydxy=2xcosx;y=x4xcosttdt In Problems 2528 use (12) to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution. 27. x2dydx+xy=10sinx;y=5x+10x1xsinttdt In Problems 2528 use (12) to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution. 28. dydx+2xy=1;y=ex2+ex20xet2dt Verify that the piecewise-defined function y={x2,x0x2,x0 is a solution of the differential equation xy 2y = 0 on (, ).In Example 7 we saw that y=1(x)=25x2 and y=2(x)=25x2 are solutions of dy/dx = x/y on the interval (5, 5). Explain why the piecewise-defined function y={25x2,5x025x2,0x5 is not a solution of the differential equation on the interval (5, 5).In Problems 31-34 find values of m so that the function y = emx is a solution of the given differential equation. 31. y + 2y = 0 In Problems 31-34 find values of m so that the function y = emx is a solution of the given differential equation. 32. 5y = 2y In Problems 31-34 find values of m so that the function y = emx is a solution of the given differential equation. 33. y 5y + 6y = 0 In Problems 31-34 find values of m so that the function y = emx is a solution of the given differential equation. 34. 2y + 7y 4y = 0 In Problems 35 and 36 find values of m so that the function y = xm is a solution of the given differential equation. 35. xy + 2y = 0 In Problems 35 and 36 find values of m so that the function y = xm is a solution of the given differential equation. 36. x2y 7xy + 15y = 0 In Problems 3740 use the concept that y = c, x , is a constant function if and only if y = 0 to determine whether the given differential equation possesses constant solutions. 3xy + 5y = 10 In Problems 3740 use the concept that y = c, x , is a constant function if and only if y = 0 to determine whether the given differential equation possesses constant solutions. y = y2 + 2y 3 In Problems 3740 use the concept that y = c, x , is a constant function if and only if y = 0 to determine whether the given differential equation possesses constant solutions. (y 1)y = 1 In Problems 3740 use the concept that y = c, x , is a constant function if and only if y = 0 to determine whether the given differential equation possesses constant solutions. y + 4y + 6y = 10 41E In Problems 41 and 42 verify that the indicated pair of functions is a solution of the given system of differential equations on the interval (, ). 42. d2xdt2=4y+et d2ydt2=4xet; x = cos 2t + sin 2t + 15et, y = cos 2t sin 2t 15et 43E Make up a differential equation that you feel confident possesses only the trivial solution y = 0. Explain your reasoning.What function do you know from calculus is such that its first derivative is itself? Its first derivative is a constant multiple k of itself? Write each answer in the form of a first-order differential equation with a solution.What function (or functions) do you know from calculus is such that its second derivative is itself? Its second derivative is the negative of itself? Write each answer in the form of a secondorder differential equation with a solution.The function y = sin x is an explicit solution of the first-order differential equation dydx=1y2. Find an interval I of definition. [Hint: I is not the interval (, ).]Discuss why it makes intuitive sense to presume that the linear differential equation y + 2y + 4y = 5 sin t has a solution of the form y = A sin t + B cos t, where A and B are constants. Then find specific constants A and B so that y = A sin t = B cos t is a particular solution of the DE.In Problems 49 and 50 the given figure represents the graph of an implicit solution G(x, y) = 0 of a differential equation dy/dx = f(x, y). In each ease the relation G(x, y) = 0 implicitly defines several solutions of the DE. Carefully reproduce each figure on a piece of paper. Use different colored pencils to mark off segments, or pieces, on each graph that correspond to graphs of solutions. Keep in mind that a solution must be a function and differentiable. Use the solution curve to estimate an interval I of definition of each solution . 49. FIGURE 1.1.6 Graph for Problem 49 In Problems 49 and 50 the given figure represents the graph of an implicit solution G(x, y) = 0 of a differential equation dy/dx = f(x, y). In each case the relation G(x, y) = 0 implicitly defines several solutions of the DE. Carefully reproduce each figure on a piece of paper. Use different colored pencils to mark off segments, or pieces, on each graph that correspond to graphs of solutions. Keep in mind that a solution must be a function and differentiable. Use the solution curve to estimate an interval I of definition of each solution . FIGURE 1.1.7 Graph for Problem 50 The graphs of members of the one-parameter family x3 + y3 = 3cxy are called folia of Descartes. Verify that this family is an implicit solution of the first-order differential equation dydx=y(y32x3)x(2y3x3).52E In Example 7 the largest interval I over which the explicit solutions y = 1(x) and y = 2(x) are defined is the open interval (5, 5). Why cant the interval I of definition be the closed interval [5, 5]?In Problem 21 a one-parameter family of solutions of the DE P = P(1 P) is given. Does any solution curve pass through the point (0, 3)? Through the point (0, 1)?Discuss, and illustrate with examples, how to solve differential equations of the forms dy/dx = f(x) and d2y/dx2 = f(x).The differential equation x(y)2 4y 12x3 = 0 has the form given in (4). Determine whether the equation can be put into the normal form dy/dx =f(x, y).57E Find a linear second-order differential equation F(x, y, y, y) = 0 for which y = c1x + c2x2 is a two-parameter family of solutions. Make sure that your equation is free of the arbitrary parameters c1 and c2.59E 60E Consider the differential equation dy/dx = y(a by), where a and b are positive constants. (a) Either by inspection or by the method suggested in Problems 3740, find two constant solutions of the DE. (b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution y = (x) is increasing. Find intervals on which y = (x) is decreasing. (c) Using only the differential equation, explain why y = a/2b is the y-coordinate of a point of inflection of the graph of a nonconstant solution y = (x). (d) On the same coordinate axes, sketch the graphs of the two constant solutions found in part (a). These constant solutions partition the xy-plane into three regions. In each region, sketch the graph of a nonconstant solution y = (x) whose shape is suggested by the results in parts (b) and (c).Consider the differential equation y = y2 + 4. (a) Explain why there exist no constant solutions of the DE. (b) Describe the graph of a solution y = (x). For example, can a solution curve have any relative extrema? (c) Explain why y = 0 is the y-coordinate of a point of inflection of a solution curve. (d) Sketch the graph of a solution y = (x) of the differential equation whose shape is suggested by parts (a)(c).In Problems 1 and 2, y = 1/(1 + c1ex) is a one-parameter family of solutions of the first-order DE y = y y2. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. y(0)=13 In Problems 1 and 2, y = 1/(1 + c1ex) is a one-parameter family of solutions of the first-order DE y = y y2. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. y(1) = 2 In Problems 36, y = 1/(x2 + c) is a one-parameter family of solutions of the first-order DE y + 2xy2 = 0. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined. 3. y(2)=13 In Problems 36, y = 1/(x2 + c) is a one-parameter family of solutions of the first-order DE y + 2xy2 = 0. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined. 4. y(2)=12 In Problems 36, y = 1/(x2 + c) is a one-parameter family of solutions of the first-order DE y + 2xy2 = 0. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined. 5. y(0) = 1 In Problems 36, y = 1/(x2 + c) is a one-parameter family of solutions of the first-order DE y + 2xy2 = 0. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined. 6. y(12)=4 In Problems 710, x = c1 cos t + c2 sin t is a two-parameter family of solutions of the second-order DE x + x = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 7. x(0) = 1, x(0) = 8 8E In Problems 710, x = c1 cos t + c2 sin t is a two-parameter family of solutions of the second-order DE x + x = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 9. x(/6)=12,x(/6)=0 In Problems 710, x = c1 cos t + c2 sin t is a two-parameter family of solutions of the second-order DE x + x = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 10. x(/4)=2,x(/4)=22 In Problems 1114, y = c1ex + c2ex is a two-parameter family of solutions of the second-order DE y y = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 11. y(0)=1y(0)=2 In Problems 1114, y = c1ex + c2ex is a two-parameter family of solutions of the second-order DE y y = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 12. y(1)=0y(1)=e In Problems 1114, y = c1ex + c2ex is a two-parameter family of solutions of the second-order DE y y = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 13. y(1)=5y(1)=5 In Problems 1114, y = c1ex + c2ex is a two-parameter family of solutions of the second-order DE y y = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 14. y(0)=0y(0)=0 In Problems 15 and 16 determine by inspection at least two solutions of the given first-order IVP. 15. y=3y2/3,y(0)=0 In Problems 15 and 16 determine by inspection at least two solutions of the given first-order IVP. 16. xy=2y,y(0)=0 In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region. 17. dydx=y2/3 In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region. 18. dydx=xy In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region. 19. xdydx=y In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region. 20. dydxy=x In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region. 21. (4y2)y=x2 In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region. 22. (1+y3)y=x2 In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region. 23. (x2 + y2)y = y2 In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region. 24. (y x)y = y + x In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation y=y29 possesses a unique solution through the given point. 25. (1, 4)In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation y=y29 possesses a unique solution through the given point. 26. (5, 3)In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation y=y29 possesses a unique solution through the given point. 27. (2, 3)In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation y=y29 possesses a unique solution through the given point. 28. (1, 1)(a) By inspection find a one-parameter family of solutions of the differential equation xy = y. Verify that each member of the family is a solution of the initial-value problem xy = y, y(0) = 0. (b) Explain part (a) by determining a region R in the xy-plane for which the differential equation xy = y would have a unique solution through a point (x0, y0) in R. (c) Verify that the piecewise-defined function y={0,x0x,x0 (a) Verify that y = tan (x + c) is a one-parameter family of solutions of the differential equation y = 1 + y2. (b) Since f(x, y) = 1 + y2 and f/y = 2y are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Use the family of solutions in part (a) to find an explicit solution of the first-order initial-value problem y = 1 + y2, y(0) = 0. Even though x0 = 0 is in the interval (2, 2), explain why the solution is not defined on this interval. (c) Determine the largest interval I of definition for the solution of the initial-value problem in part (b).(a) Verify that y = 1 /(x + c) is a one-parameter family of solutions of the differential equation y = y2. (b) Since f(x, y) = y2 and f/y = 2y are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Find a solution from the family in part (a) that satisfies y(0) = 1. Then find a solution from the family in part (a) that satisfies y(0) = 1. Determine the largest interval I of definition for the solution of each initial-value problem. (c) Determine the largest interval I of definition for the solution of the first-order initial-value problem y = y2, y(0) = 0. [Hint: The solution is not a member of the family of solutions in part (a).](a) Show that a solution from the family in part (a) of Problem 31 that satisfies y = y2, y(1) = 1, is y = 1/(2 x). (b) Then show that a solution from the family in part (a) of Problem 31 that satisfies y = y2, y(3) = 1, is y = 1/(2 x). (c) Are the solutions in parts (a) and (b) the same?(a) Verify that 3x2 y2 = c is a one-parameter family of solutions of the differential equation y dy/dx = 3x. (b) By hand, sketch the graph of the implicit solution 3x2 y2 = 3. Find all explicit solutions y = (x) of the DE in part (a) defined by this relation. Give the interval I of definition of each explicit solution. (c) The point (2, 3) is on the graph of 3x y2 = 3, but which of the explicit solutions in part (b) satisfies y(2) = 3?(a) Use the family of solutions in part (a) of Problem 33 to find an implicit solution of the initial-value problem y dy/dx = 3x, y(2) = 4. Then, by hand, sketch the graph of the explicit solution of this problem and give its interval I of definition. (b) Are there any explicit solutions of y dy/dx = 3x that pass through the origin?In Problems 3538 the graph of a member of a family of solutions of a second-order differential equation d2y/dx2 = f(x, y, y) is given. Match the solution curse with at least one pair of the following initial conditions. (a) y(1) = 1, y(1) = 2 (b) y(1) = 0, y(1) = 4 (c) y(1) = 1, y(1) = 2 (d) y(0) = 1, y(0) = 2 (e) y(0) = 1, y(0) = 0 (f) y(0) = 4, y(0) = 2 35. FIGURE 1.2.7 Graph for Problem 35 In Problems 3538 the graph of a member of a family of solutions of a second-order differential equation d2y/dx2 = f(x, y, y) is given. Match the solution curve with at least one pair of the following initial conditions. (a) y(1) = 1, y(1) = 2 (b) y(1) = 0, y(1) = 4 (c) y(1) = 1, y(1) = 2 (d) y(0) = 1, y(0) = 2 (e) y(0) = 1, y(0) = 0 (f) y(0) = 4, y(0) = 2 36. FIGURE 1.2.8 Graph for Problem 36 In Problems 3538 the graph of a member of a family of solutions of a second-order differential equation d2y/dx2 = f(x, y, y) is given. Match the solution curve with at least one pair of the following initial conditions. (a) y(1) = 1, y(1) = 2 (b) y(1) = 0, y(1) = 4 (c) y(1) = 1, y(1) = 2 (d) y(0) = 1, y(0) = 2 (e) y(0) = 1, y(0) = 0 (f) y(0) = 4, y(0) = 2 37. FIGURE 1.2.9 Graph for Problem 37 In Problems 3538 the graph of a member of a family of solutions of a second-order differential equation d2y/dx2 = f(x, y, y) is given. Match the solution curve with at least one pair of the following initial conditions. (a) y(1) = 1, y(1) = 2 (b) y(1) = 0, y(1) = 4 (c) y(1) = 1, y(1) = 2 (d) y(0) = 1, y(0) = 2 (e) y(0) = 1, y(0) = 0 (f) y(0) = 4, y(0) = 2 38. FIGURE 1.2.10 Graph for Problem 38 In Problems 3944, y = c1 cos 2x + c2 sin 2x is a two-parameter family of solutions of the second-order DE y + 4y = 0. If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. 39. y(0) = 0, y(/4) = 3 In Problems 3944, y = c1 cos 2x + c2 sin 2x is a two-parameter family of solutions of the second-order DE y + 4y = 0. If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. 40. y(0) = 0, y() = 0 In Problems 3944, y = c1 cos 2x + c2 sin 2x is a two-parameter family of solutions of the second-order DE y + 4y = 0. If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. 41. y(0) = 0, y(/6) = 0 In Problems 3944, y = c1 cos 2x + c2 sin 2x is a two-parameter family of solutions of the second-order DE y + 4y = 0. If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. 42. y(0) = 1, y() = 5 In Problems 3944, y = c1 cos 2x + c2 sin 2x is a two-parameter family of solutions of the second-order DE y + 4y = 0. If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. 43. y(0) = 0, y() = 2 In Problems 3944, y = c1 cos 2x + c2 sin 2x is a two-parameter family of solutions of the second-order DE y + 4y = 0. If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. 44. y(/2) = 1, y() = 0 Find a function whose graph at each point (x, y) has the slope given by 8e2x + 6x and has the y-intercept (0, 9).46E Consider the initial-value problem y = x 2y, y(0) = 12. Determine which of the two curves shown in Figure 1.2.11 is the only plausible solution curve. Explain your reasoning. FIGURE 1.2.11 Graphs for Problem 47 Show that x=0y1t3+1dt is an implicit solution of the initial-value problem 2d2ydx23y2=0, y(0) = 0, y(0) = 1. Assume that y 0. [Hint: The integral is nonelementary. See (ii) in the Remarks at the end of Section 1.1.]49E Suppose that the first-order differential equation dy/dx = f(x, y) possesses a one-parameter family of solutions and that f(x, y) satisfies the hypotheses of Theorem 1.2.1 in some rectangular region R of the xy-plane. Explain why two different solution curves cannot intersect or be tangent to each other at a point (x0, y0) in R.The functions y(x)=116x4, x and y(x)={0,x0116x4,x0 have the same domain but are clearly different. See Figures 1.2.12(a) and 1.2.12(b), respectively. Show that both functions are solutions of the initial-value problem dy/dx = xy1/2, y(2) = 1 on the interval (, ). Resolve the apparent contradiction between this fact and the last sentence in Example 5. FIGURE 1.2.12 Two solutions of the IVP in Problem 51 Under the same assumptions that underlie the model in (1), determine a differential equation for the population P(t) of a country when individuals are allowed to immigrate into the country at a constant rate r 0. What is the differential equation for the population P(t) of the country when individuals are allowed to emigrate from the country at a constant rate r 0?The population model given in (1) fails to take death into consideration; the growth rate equals the birth rate. In another model of a changing population of a community it is assumed that the rate at which the population changes is a net ratethat is, the difference between the rate of births and the rate of deaths in the community. Determine a model for the population P(t) if both the birth rate and the death rate are proportional to the population present at time t 0.Using the concept of net rate introduced in Problem 2, determine a model for a population P(t) if the birth rate is proportional to the population present at time t but the death rate is proportional to the square of the population present at time t.Modify the model in Problem 3 for net rate at which the population P(t) of a certain kind of fish changes by also assuming that the fish are harvested at a constant rate h 0.A cup of coffee cools according to Newtons law of cooling (3). Use data from the graph of the temperature T(t) in Figure 1.3.10 to estimate the constants Tm, T0, and k in a model of the form of a first-order initial-value problem: dT/dt = k(T Tm), T(0) = T0.The ambient temperature Tm in (3) could be a function of time t. Suppose that in an artificially controlled environment, Tm(t) is periodic with a 24-hour period, as illustrated in Figure 1.3.11. Devise a mathematical model for the temperature T(t) of a body within this environment.Suppose a student carrying a flu virus returns to an isolated college campus of 1000 students. Determine a differential equation for the number of people x(t) who have contracted the flu if the rate at which the disease spreads is proportional to the number of interactions between the number of students who have the flu and the number of students who have not yet been exposed to it.At a time denoted as t = 0 a technological innovation is introduced into a community that has a fixed population of n people. Determine a differential equation for the number of people x(t) who have adopted the innovation at time t if it is assumed that the rate at which the innovations spread through the community is jointly proportional to the number of people who have adopted it and the number of people who have not adopted it.Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt have been dissolved. Pure water is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at the same rate. Determine a differential equation for the amount of salt A(t) in the tank at time t 0. What is A(0)?Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at a slower rate of 2 gal/min. If the concentration of the solution entering is 2 lb/gal, determine a differential equation for the amount of salt A(t) in the tank at time t 0.What is the differential equation in Problem 10, if the well-stirred solution is pumped out at a faster rate of 3.5 gal/min?Generalize the model given in equation (8) of this section by assuming that the large tank initially contains N0 number of gallons of brine, rin and rout are the input and output rates of the brine, respectively (measured in gallons per minute), cin is the concentration of the salt in the inflow, c(t) the concentration of the salt in the tank as well as in the outflow at time t (measured in pounds of salt per gallon), and A(t) is the amount of salt in the tank at time t 0.Suppose water is leaking from a tank through a circular hole of area Ah at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to cAh2gh, where c (0 c 1) is an empirical constant. Determine a differential equation for the height h of water at time t for the cubical tank shown in Figure 1.3.12. The radius of the hole is 2 in., and g = 32 ft/s2.The right-circular conical tank shown in Figure 1.3.13 loses water out of a circular hole at its bottom. Determine a differential equation for the height of the water h at time t 0. The radius of the hole is 2 in., g = 32 ft/s2, and the friction/ contraction factor introduced in Problem 13 is c = 0.6.A series circuit contains a resistor and an inductor as shown in Figure 1.3.14. Determine a differential equation for the current i(t) if the resistance is R, the inductance is L, and the impressed voltage is E(t).A series circuit contains a resistor and a capacitor as shown in Figure 1.3.15. Determine a differential equation for the charge q(t) on the capacitor if the resistance is R, the capacitance is C, and the impressed voltage is E(t). FIGURE 1.3.15 RC-series circuit in Problem 16 For high-speed motion through the airsuch as the skydiver shown in Figure 1.3.16, falling before the parachute is openedair resistance is closer to a power of the instantaneous velocity v(t). Determine a differential equation for the velocity v(t) of a falling body of mass m if air resistance is proportional to the square of the instantaneous velocity. Assume the downward direction is positive. FIGURE 1.3.16 Air resistance proportional to square of velocity in Problem 17 A cylindrical barrel s feet in diameter of weight w lb is floating in water as shown in Figure 1.3.17(a). After an initial depression the barrel exhibits an up-and-down bobbing motion along a vertical line. Using Figure 1.3.17(b), determine a differential equation for the vertical displacement y(t) if the origin is taken to be on the vertical axis at the surface of the water when the barrel is at rest. Use Archimedes principle: Buoyancy, or upward force of the water on the barrel, is equal to the weight of the water displaced. Assume that the downward direction is positive, that the weight density of water is 62.4 lb/ft3, and that there is no resistance between the barrel and the water. FIGURE 1.3.17 Bobbing motion of floating barrel in Problem 18 After a mass m is attached to a spring, it stretches it s units and then hangs at rest in the equilibrium position as shown in Figure 1.3.18(b). After the spring/mass system has been set in motion, let x(t) denote the directed distance of the mass beyond the equilibrium position. As indicated in Figure 1.3.18(c), assume that the downward direction is positive, that the motion takes place in a vertical straight line through the center of gravity of the mass, and that the only forces acting on the system are the weight of the mass and the restoring force of the stretched spring. Use Hookes law: The restoring force of a spring is proportional to its total elongation. Determine a differential equation for the displacement x(t) at time t 0. FIGURE 1.3.18 Spring/mass system in Problem 19 In Problem 19, what is a differential equation for the displacement x(t) if the motion takes place in a medium that imparts a damping force on the spring/mass system that is proportional to the instantaneous velocity of the mass and acts in a direction opposite to that of motion?A small single-stage rocket is launched vertically as shown in Figure 1.3.19. Once launched, the rocket consumes its FIGURE 1.3.19 Single stage rocket in Problem 21 fuel, and so its total mass m(t) varies with time t 0. If it is assumed that the positive direction is upward, air resistance is proportional to the instantaneous velocity v of the rocket, and R is the upward thrust or force generated by the propulsion system, then construct a mathematical model for the velocity v(t) of the rocket. [Hint: See (14) in Section 1.3.]In Problem 21, the mass m(t) is the sum of three different masses: m(t) = mp + mv + mf(t), where mp is the constant mass of the payload, mv is the constant mass of the vehicle, and mf(t) is the variable amount of fuel. (a) Show that the rate at which the total mass m(t) of the rocket changes is the same as the rate at which the mass mf(t) of the fuel changes. (b) If the rocket consumes its fuel at a constant rate , find m(t). Then rewrite the differential equation in Problem 21 in terms of and the initial total mass m(0) = m0. (c) Under the assumption in part (b), show that the burnout time tb 0 of the rocket, or the time at which all the fuel is consumed, is tb = mf(0)/, where mf(0) is the initial mass of the fuel.By Newtons universal law of gravitation the free-fall acceleration a of a body, such as the satellite shown in Figure 1.3.20, falling a great distance to the surface is not the constant g. Rather, the acceleration a is inversely proportional to the square of the distance from the center of the Earth, a = k/r2, where k is the constant of proportionality. Use the fact that at the surface of the Earth r = R and a = g to determine k. If the positive direction is upward, use Newtons second law and his universal law of gravitation to find a differential equation for the distance r. FIGURE 1.3.20 Satellite in Problem 23 Suppose a hole is drilled through the center of the Earth and a bowling ball of mass m is dropped into the hole, as shown in Figure 1.3.21. Construct a mathematical model that describes the FIGURE 1.3.21 Hole through Earth in Problem 24 motion of the ball. At time t let r denote the distance from the center of the Earth to the mass m, M denote the mass of the Earth, Mr denote the mass of that portion of the Earth within a sphere of radius r, and denote the constant density of the Earth.Learning Theory In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose M denotes the total amount of a subject to be memorized and A(t) is the amount memorized in time t 0. Determine a differential equation for the amount A(t).Forgetfulness In Problem 25 assume that the rate at which material is forgotten is proportional to the amount memorized in time t 0. Determine a differential equation for the amount A(t) when forgetfulness is taken into account.Infusion of a Drug A drug is infused into a patients bloodstream at a constant rate of r grams per second. Simultaneously, the drug is removed at a rate proportional to the amount x(t) of the drug present at time t. Determine a differential equation for the amount x(t).Tractrix A motorboat starts at the origin and moves in the direction of the positive x-axis, pulling a waterskier along a curve C called a tractrix. See Figure 1.3.22. The waterskier, initially located on the y-axis at the point (0, a), is pulled by a rope of constant length a that is kept taut throughout the motion. At time t 0 the waterskier is at point P(x, y). Assume that the rope is always tangent to C. Use the concept of slope to determine a differential equation for the path C of motion. FIGURE 1.3.22 Waterskier in Problem 28 Reflecting surface Assume that when the plane curve C shown in Figure 1.3.23 is revolved about the x-axis, it generates FIGURE 1.3.23 Reflecting surface in Problem 29 a surface of revolution with the property that all light rays L parallel to the x-axis striking the surface are reflected to a single point O (the origin). Use the fact that the angle of incidence is equal to the angle of reflection to determine a differential equation that describes the shape of the curve C. Such a curve C is important in applications ranging from construction of telescopes to satellite antennas, automobile headlights, and solar collectors. [Hint: Inspection of the figure shows that we can write = 2. Why? Now use an appropriate trigonometric identity.]Reread Problem 45 in Exercises 1.1 and then give an explicit solution P(t) for equation (1). Find a one-parameter family of solutions of (1).31E 32E 33E Rotating Fluid As shown in Figure 1.3.24(a), a right-circular cylinder partially filled with fluid is rotated with a constant angular velocity about a vertical y-axis through its center. The rotating fluid forms a surface of revolution S. To identify S, we first establish a coordinate system consisting of a vertical plane determined by the y-axis and an x-axis drawn perpendicular to the y-axis such that the point of intersection of the axes (the origin) is located at the lowest point on the surface S. We then seek a function y = f(x) that represents the curve C of intersection of the surface S and the vertical coordinate plane. Let the point P(x, y) denote the position of a particle of the rotating fluid of mass m in the coordinate plane. See Figure 1.3.24(b). (a) At P there is a reaction force of magnitude F due to the other particles of the fluid which is normal to the surface S. By Newtons second law the magnitude of the net force acting on the particle is m2x. What is this force? Use Figure 1.3.24(b) to discuss the nature and origin of the equations Fcos=mg,Fsin=m2x. (b) Use part (a) to find a first-order differential equation that defines the function y = f(x).35E 36E Let It snow The snowplow problem is a classic and appears in many differential equations texts, but it was probably made famous by Ralph Palmer Agnew: One day it started snowing at a heavy and steady rate. A snowplow started out at noon, going 2 miles the first hour and 1 mile the second hour. What time did it start snowing? Find the textbook Differential Equations, Ralph Palmer Agnew, McGraw-Hill Book Co., and then discuss the construction and solution of the mathematical model.Population Dynamics Suppose that dP/dt = 0.15 P(t) represents a mathematical model for the growth of a certain cell culture, where P(t) is the size of the culture (measured in millions of cells) at time t 0 (measured in hours). How fast is the culture growing at the time when the size of the culture reaches 2 million cells?Radioactive Decay Suppose that dA/dt = 0.0004332 A(t) represents a mathematical model for the decay of radium-226, where A(t) is the amount of radium (measured in grams) remaining at time t 0 (measured in years). How much of the radium sample remains at the time when the sample is decaying at a rate of 0.002 grams per year?Reread this section and classify each mathematical model linear or nonlinear.In Problems 1 and 2 fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol c1 and has the form dy/dx = f(x, y). The symbol c1 represents a constant. ddxc1e10x=In Problems 1 and 2 fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol c1 and has the form dy/dx = f(x, y). The symbol c1 represents a constant. ddx(5+c1e2x)=In Problems 3 and 4 fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y ) = 0. The symbols c1, c2, and k represent constants. 3. d2dx2(c1coskx+c2sinkx)=In Problems 3 and 4 fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y ) = 0. The symbols c1, c2, and k represent constants. 4. d2dx2(c1coshkx+c2sinhkx)=In Problems 5 and 6 compute y and y and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y y) = 0. The symbols c1 and c2 represent constants. 5. y = c1ex + c2xex In Problems 5 and 6 compute y and y and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y y) = 0. The symbols c1 and c2 represent constants. 6. y = c1ex cos x + c2xex sin x In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y = 0, (b) y = 2, (c) y = 2x, (d) y = 2x2. 7. xy = 2y In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y = 0, (b) y = 2, (c) y = 2x, (d) y = 2x2. 8. y = 2 In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y = 0, (b) y = 2, (c) y = 2x, (d) y = 2x2. 9. y = 2y 4 In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y = 0, (b) y = 2, (c) y = 2x, (d) y = 2x2. 10. xy = y In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y = 0, (b) y = 2, (c) y = 2x, (d) y = 2x2 11. y + 9y = 18 In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y = 0, (b) y = 2, (c) y = 2x, (d) y = 2x2. 12. xy y = 0 In Problems 13 and 14 determine by inspection at least one solution of the given differential equation. 13. y = y In Problems 13 and 14 determine by inspection at least one solution of the given differential equation. 14. y = y(y 3)In Problems 15 and 16 interpret each statement as a differential equation. 15. On the graph of y = (x) the slope of the tangent line at a point P(x, y) is the square of the distance from P(x, y) to the origin.In Problems 15 and 16 interpret each statement as a differential equation. 16. On the graph of y = (x) the rate at which the slope changes with respect to x at a point P(x, y) is the negative of the slope of the tangent line at P(x, y).(a) Give the domain of the function y = x2/3. (b) Give the largest interval I of definition over which y = x2/3 is a solution of the differential equation 3xy 2y = 0.(a) Verify that the one-parameter family y2 2y = x2 x + c is an implicit solution of the differential equation (2y 2)y = 2x 1. (b) Find a member of the one-parameter family in part (a) that satisfies the initial condition y(0) = 1. (c) Use your result in part (b) to find an explicit function y = (x) that satisfies y(0) = 1. Give the domain of the function . Is y = (x) a solution of the initial-value problem? If so, give its interval I of definition; if not, explain.The function y = x 2/x is a solution of the DE xy + y = 2x. Find x0 and the largest interval I for which y(x) is a solution of the first-order IVP xy + y = 2x, y(x0) = 1.Suppose that y(x) denotes a solution of the first-order IVP y = x2 + y2, y(1) = 1 and that y(x) possesses at least a second derivative at x = 1. In some neighborhood of x = 1 use the DE to determine whether y(x) is increasing or decreasing and whether the graph y(x) is concave up or concave down.A differential equation may possess more than one family of solutions. (a) Plot different members of the families y = 1(x) = x2 + c1 and y = 2(x) = x2 + c2. (b) Verify that y = 1(x) and y = 2(x) are two solutions of the nonlinear first-order differential equation (y)2 = 4x2. (c) Construct a piecewise-defined function that is a solution of the nonlinear DE in part (b) but is not a member of either family of solutions in part (a).What is the slope of the tangent line to the graph of a solution of y=6y+5x3 that passes through (1, 4)?In Problems 2326 verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition I for each solution. 23.y+y=2cosx2sinx;y=xsinx+xcosx In Problems 2326 verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition I for each solution. 24.y+y=secx;y=xsinx+(cosx)ln(cosx)In Problems 2326 verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition I for each solution. 25.x2y+xy+y=0;y=son(lnx)In Problems 2326 verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition I for each solution. 26.x2y+xy+y=sec(lnx)y=cos(lnx)In(cos(Inx))+(Inx)sin(Inx)In Problems 2730 use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution. 27.dydx+(sinx)y=x;y=ecosx0xtecostdt In Problems 2730 use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution. 28.dydx2xy=ex;y=ex20xett2dt In Problems 2730 use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution. 29. x2y+(x2x)y+(1x)y=0;y=x1xe1tdt In Problems 2730 use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution. 30. y+y=ex2;y=sinx0xet2costdtcosx0xet2sintdt In Problems 3134 verify that the indicated expression is an implicit solution of the given differential equation. 31. xdydx+y=1y2;x3y3=x3+1 In Problems 3134 verify that the indicated expression is an implicit solution of the given differential equation. 32. (dydx)2+1=1y2;(x5)2+y2=1 In Problems 3134 verify that the indicated expression is an implicit solution of the given differential equation. 33. y=2y(y)3;y3+3y=13x In Problems 3134 verify that the indicated expression is an implicit solution of the given differential equation. 34. (1xy)y=y2;y=exy In Problems 3538, y = c1e3x + c2ex 2x is a two-parameter family of the second-order DE y 2y 3y = 6x + 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 35. y (0) = 0, y(0) = 0 In Problems 3538, y = c1e3x + c2ex 2x is a two-parameter family of the second-order DE y 2y 3y = 6x + 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 36. y (0) = 1, y(0) = 3 In Problems 3538, y = c1e3x + c2ex 2x is a two-parameter family of the second-order DE y 2y 3y = 6x + 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 37. y (1) = 4, y(1) = 2 In Problems 3538, y = c1e3x + c2ex 2x is a two-parameter family of the second-order DE y 2y 3y = 6x + 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 38. y (1) = 0, y(1) = 1 The graph of a solution of a second-order initial-value problem d2y/dx2 = f(x, y, y), y(2) = y0, y(2) = y1, is given in Figure 1.R.1. Use the graph to estimate the values of y0 and y1. FIGURE 1.R.1 Graph for Problem 39 A tank in the form of a right-circular cylinder of radius 2 feet and height 10 feet is standing on end. If the tank is initially full of water and water leaks from a circular hole of radius 12 inch at its bottom, determine a differential equation for the height h of the water at time t 0. Ignore friction and contraction of water at the hole.In Problems 14 reproduce the given computer-generated direction field. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for each solution curve. 1. dydx=x2y2 (a) y(2) = 1 (b) y(3) = 0 (c) y(0) = 2 (d) y(0) = 0 FIGURE 2.1.12 Direction field for Problem 1 In Problems 14 reproduce the given computer-generated direction field. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for each solution curve. 2. dydx=e0.01xy2 (a) y(6) = 0 (b) y(0) = 1 (c) y(0) = 4 (d) y(8) = 4 FIGURE 2.1.13 Direction field for Problem 2 dydx=1xy (a) y(0) = 0 (b) y(1) = 0 (c) y(2) = 2 (d) y(0) = 4 FIGURE 2.1.14 Direction field for Problem 3 In Problems 14 reproduce the given computer-generated direction field. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for each solution curve. 4. dydx=(sinx)cosy (a) y(0) = 1 (b) y(1) = 0 (c) y(3) = 3 (d) y(0)=52 FIGURE 2.1.15 Direction field for Problem 4 In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. 5. y = x (a) y(0) = 0 (b) y(0) = 3 In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. 6. y = x + y (a) y(2) = 2 (b) y(1) = 3 In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. 7. ydydx=x (a) y(1) = 1 (b) y(0) = 4 In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. 8. dydx=1y (a) y(0) = 1 (b) y(2) = 1 In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. 9. dydx=0.2x2+y (a) y(0)=12 (b) y(2) = 1 In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. 10. dydx=xey (a) y(0) = 2 (b) y(1) = 2.5 In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. 11. y=ycos2x (a) y(2) = 2 (b) y(1) = 0 In Problems 512 use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. 12. dydx=1yx (a) y(12)=2 (b) y(32)=0 In Problems 13 and 14 the given figure represents the graph of f(y) and f(x), respectively. By hand, sketch a direction field over an appropriate grid for dy/dx = f(y) (Problem 13) and then for dy/dx = f(x) (Problem 14). 13. FIGURE 2.1.16 Graph for Problem 13 In Problems 13 and 14 the given figure represents the graph of f(y) and f(x), respectively. By hand, sketch a direction field over an appropriate grid for dy/dx = f(y) (Problem 13) and then for dy/dx = f(x) (Problem 14). 14. Figure 2.1.17 Graph for Problem 14 In parts (a) and (b) sketch isoclines f(x, y) = c (see the Remarks on page 39) for the given differential equation using the indicated values of c. Construct a direction field over a grid by carefully drawing lineal elements with the appropriate slope at chosen points on each isocline. In each case, use this rough direction field to sketch an approximate solution curve for the IVP consisting of the DE and the initial condition y(0) = 1. (a) dy/dx = x + y; c an integer satisfying 5 c 5 (b) dy/dx = x2 + y2; c=14, c = 1, c=94, c = 4 (a) Consider the direction field of the differential equation dy/dx = x(y 4)2 2, but do not use technology to obtain it. Describe the slopes of the lineal elements on the lines x = 0, y = 3, y = 4, and y = 5. (b) Consider the IVP dy/dx = x(y 4)2 2, y(0) = y0, where y0 4. Can a solution y(x) as x ? Based on the information in part (a), discuss.Consider the autonomous first-order differential equation dy/dx = y y3 and the initial condition y(0) = y0. By hand, sketch the graph of a typical solution y(x) when y0 has the given values. (a) y0 1 (b) 0 y0 1 (c) 1 y0 0 (d) y0 1 Consider the autonomous first-order differential equation dy/dx = y2 y4 and the initial condition y(0) = y0. By hand, sketch the graph of a typical solution y(x) when y0 has the given values. (a) y0 1 (b) 0 y0 1 (c) 1 y0 0 (d) y0 1 In Problems 21-28 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions. 21. dydx=y23y In Problems 21-28 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions. 22. dydx=y2y3 In Problems 21-28 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions. 23. dydx=(y2)4 In Problems 21-28 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions. 24. dydx=10+3yy2 In Problems 21-28 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions. 25. dydx=y2(4y2)In Problems 21-28 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions. 26. dydx=y(2y)(4y)In Problems 21-28 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions. 27. dydx=yln(y+2)28E In Problems 29 and 30 consider the autonomous differential equation dy/dx =f(y), where the graph of f is given. Use the graph to locate the critical points of each differential equation. Sketch a phase portrait of each differential equation. By hand, sketch typical solution curves in the subregions in the xy-plane determined by the graphs of the equilibrium solutions. 29. FIGURE 2.1.18 Graph for Problem 29 In Problems 29 and 30 consider the autonomous differential equation dy/dx = f(y), where the graph of f is given. Use the graph to locate the critical points of each differential equation. Sketch a phase portrait of each differential equation. By hand, sketch typical solution curves in the subregions in the xy-plane determined by the graphs of the equilibrium solutions. FIGURE 2.1.19 Graph for Problem 30 Consider the autonomous DE dy/dx = (2/)y sin y Determine the critical points of the equation. Discuss a way of obtaining a phase portrait of the equation. Classify the critical points as asymptotically stable, unstable, or semi-stable.A critical point c of an autonomous first-order DE is said to be isolated if there exists some open interval that contains c but no other critical point. Can there exist an autonomous DE of the form given in (2) for which every critical point is nonisolated? Discuss; do not think profound thoughts.Suppose that y(x) is a nonconstant solution of the autonomous equation dy/dx = f(y) and that c is a critical point of the DE. Discuss: Why cant the graph of y(x) cross the graph of the equilibrium solution y = c? Why cant f(y) change signs in one of the subregions discussed on page 40? Why cant y(x) be oscillatory or have a relative extremum (maximum or minimum)?34E Using the autonomous equation (2), discuss how it is possible to obtain information about the location of points of inflection of a solution curve.36E Suppose the autonomous DE in (2) has no critical points. Discuss the behavior of the solutions.Population Model The differential equation in Example 3 is a well-known population model. Suppose the DE is changed to dPdt=P(aPb), where a and b are positive constants. Discuss what happens to the population P as time t increases.Population Model Another population model is given by dpdt=kPh, where h and k are positive constants. For what initial values P(0) = P0 does this model predict that the population will go extinct?Terminal Velocity In Section 1.3 we saw that the autonomous differential equation mdvdt=mgkv, where k is a positive constant and g is the acceleration due to gravity, is a model for the velocity v of a body of mass m that is falling under the influence of gravity. Because the term kv represents air resistance, the velocity of a body falling from a great height does not increase without bound as time t increases. Use a phase portrait of the differential equation to find the limiting, or terminal, velocity of the body. Explain your reasoning.Suppose the model in Problem 40 is modified so that air resistance is proportional to v2, that is, mdvdt=mgkv2. See Problem 17 in Exercises 1.3. Use a phase portrait to find the terminal velocity of the body. Explain your reasoning.Chemical reactions When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled by the autonomous differential equation dXdt=k(X)(X), where k 0 is a constant of proportionality and 0. Here X(t) denotes the number of grams of the new compound formed in time t. (a) Use a phase portrait of the differential equation to predict the behavior of X(t) as t . (b) Consider the case when = . Use a phase portrait of the differential equation to predict the behavior of X(t) as t when X(0) . When X(0) . (c) Verify that an explicit solution of the DE in the case when k = 1 and = is X(t) = 1/(t + c). Find a solution that satisfies X(0) = /2. Then find a solution that satisfies X(0) = 2. Graph these two solutions. Does the behavior of the solutions as t agree with your answers to part (b)?In Problems 122 solve the given differential equation by separation of variables. dydx=sin5x In Problems 122 solve the given differential equation by separation of variables. 2. dydx=(x+1)2 In Problems 122 solve the given differential equation by separation of variables. 3. dx+e3xdy=0 In Problems 122 solve the given differential equation by separation of variables. 4. dx(y1)2dx=0 In Problems 122 solve the given differential equation by separation of variables. 5. xdydx=4y In Problems 122 solve the given differential equation by separation of variables. 6. dydx+2xy2=0 In Problems 122 solve the given differential equation by separation of variables. 7. dydx=e3x+2y In Problems 122 solve the given differential equation by separation of variables. 8. exydydx=ey+e2xy In Problems 122 solve the given differential equation by separation of variables. 9. ylnxdxdy=(y+1x)2 In Problems 122 solve the given differential equation by separation of variables. 10. dydx=(2y+34x+5)2 In Problems 122 solve the given differential equation by separation of variables. 11. cscydx+sec2xdy=0 In Problems 122 solve the given differential equation by separation of variables. 12. sin3xdx+2ycos33xdy=0 In Problems 122 solve the given differential equation by separation of variables. 13. (ey+1)2eydx+(ex+1)3exdx=0 In Problems 122 solve the given differential equation by separation of variables. 14. x(1+y2)1/2dx=y(1+x2)1/2dy In Problems 122 solve the given differential equation by separation of variables. 15. dSdr=kS In Problems 122 solve the given differential equation by separation of variables. 16. dQdt=k(Q70)In Problems 122 solve the given differential equation by separation of variables. 17. dpdt=PP2 In Problems 122 solve the given differential equation by separation of variables. 18. dNdt+N=Ntet+2 In Problems 122 solve the given differential equation by separation of variables. 19. dydx=xy+3xy3xy2x+4y8 In Problems 122 solve the given differential equation by separation of variables. 20. dydx=xy+2yx2xy3y+x3 In Problems 122 solve the given differential equation by separation of variables. 21. dydx=x1y2 In Problems 122 solve the given differential equation by separation of variables. 22. (ex+ex)dydx=y2 In Problems 2328 find an explicit solution of the given initial-value problem. 23. dxdt=4(x2+1), x(/4) = 1 In Problems 2328 find an explicit solution of the given initial-value problem. 24. dydx=y21x21, y(2) = 2 In Problems 2328 find an explicit solution of the given initial-value problem. 25. x2dydx=yxy, y(1) = 1 In Problems 2328 find an explicit solution of the given initial-value problem. 26. dydt+2y=1,y(0)=52 In Problems 2328 find an explicit solution of the given initial-value problem. 27. 1y2dx1x2dy=0,y(0)=32 In Problems 2328 find an explicit solution of the given initial-value problem. 28. (1 + x4) dy + x(1 + 4y2) dx = 0, y(1) = 0 In Problems 29 and 30 proceed as in Example 5 and find an explicit solution of the given initial-value problem. 29. dydx=yex2, y(4) = 1 In Problems 29 and 30 proceed as in Example 5 and find an explicit solution of the given initial-value problem. 30. dydx=y2sinx2,y(2)=13 In Problems 3134 find an explicit solution of the given initial-value problem. Determine the exact interval I of definition by analytical methods. Use a graphing utility to plot the graph of the solution. dydx=2x+12y,y(2)=1 In Problems 3134 find an explicit solution of the given initial-value problem. Determine the exact interval I of definition by analytical methods. Use a graphing utility to plot the graph of the solution. (2y2)dydx=3x2+4x+2, y(1) = 2 In Problems 3134 find an explicit solution of the given initial-value problem. Determine the exact interval I of definition by analytical methods. Use a graphing utility to plot the graph of the solution. ey dx ex dy = 0, y(0) = 0 In Problems 3134 find an explicit solution of the given initial-value problem. Determine the exact interval I of definition by analytical methods. Use a graphing utility to plot the graph of the solution. sin x dx + ydy = 0, y(0) = 1 (a) Find a solution of the initial-value problem consisting of the differential equation in Example 3 and each of the initial-conditions: y(0) = 2, y(0) = 2, and y(14)=1. (b) Find the solution of the differential equation in Example 3 when ln c1 is used as the constant of integration on the left-hand side in the solution and 4 ln c1 is replaced by ln c. Then solve the same initial-value problems in part (a).Find a solution of xdydx=y2y that passes through the indicated points. (a) (0, 1) (b) (0,.0) (c) (12,12) (d) (2,14)Find a singular solution of Problem 21. Of Problem 22.Show that an implicit solution of 2xsin2ydx(x2+10)cosydy=0 is given by ln(x2 + 10) + csc y = c. Find the constant solutions, if any, that were lost in the solution of the differential equation.Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems 39-42 find an explicit solution of the given initial-value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of (0, 1). 39. dydx=(y1)2, y(0) = 1 Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems 39-42 find an explicit solution of the given initial-value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of (0, 1). 40. dydx=(y1)2, y(0) = 1.01 Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems 39-42 find an explicit solution of the given initial-value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of (0, 1). 41. dydx=(y1)2+0.01, y(0) = 1 Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems 3942 find an explicit solution of the given initial-value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of (0, 1). dydx=(y1)20.01, y(0) = 1 Every autonomous first-order equation dy/dx = f(y) is separable. Find explicit solutions y1(x), y2(x), y3(x), and y4(x) of the differential equation dy/dx = y y3 that satisfy, in turn, the initial conditions y1(0) = 2,y2(0)=12, y3(0)=12 and y4(0) = 2. Use a graphing utility to plot the graphs of each solution. Compare these graphs with those predicted in Problem 19 of Exercises 2.1. Give the exact interval of definition for each solution.(a) The autonomous first-order differential equation dy/dx = 1/(y 3) has no critical points. Nevertheless, place 3 on the phase line and obtain a phase portrait of the equation. Compute d2y/dx2 to determine where solution curves are concave up and where they are concave down (see Problems 35 and 36 in Exercises 2.1). Use the phase portrait and concavity to sketch, by hand, some typical solution curves. (b) Find explicit solutions y1(x), y2(x), y3(x), and y4(x) of the differential equation in part (a) that satisfy, in turn, the initial conditions y1(0) = 4, y2(0) = 2, y3(1) = 2, and y4(1) = 4. Graph each solution and compare with your sketches in part (a). Give the exact interval of definition for each solution.In Problems 4550 use a technique of integration or a substitution to find an explicit solution of the given differential equation or initial-value problem. 45. dydx=11+sinx In Problems 4550 use a technique of integration or a substitution to find an explicit solution of the given differential equation or initial-value problem. 46. dydx=sinxy In Problems 4550 use a technique of integration or a substitution to find an explicit solution of the given differential equation or initial-value problem. 47. (x+x)dydx=y+y In Problems 4550 use a technique of integration or a substitution to find an explicit solution of the given differential equation or initial-value problem. 48. dydx=y2/3y In Problems 4550 use a technique of integration or a substitution to find an explicit solution of the given differential equation or initial-value problem. 49. dydx=exy, y(1) = 4 In Problems 4550 use a technique of integration or a substitution to find an explicit solution of the given differential equation or initial-value problem. 50. dydx=xtan1xy, y(0) = 3 51E 52E In Problems 43 and 44 we saw that every autonomous first-order differential equation dy/dx = f(y) is separable. Does this fact help in the solution of the initial-value problem dydx=1+y2sin2y,y(0)=12? Discuss. Sketch, by hand, a plausible solution curve of the problem.54E Find a function whose square plus the square of its derivative is 1.56E 57E In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 1. dydx=5y In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 2. dydx+2y=0 In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 3. dydx+y=e3x In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 4. 3dydx+12y=4 In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 5. y + 3x2y = x2 In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 6. y + 2xy = x3 In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 7. x2y + xy = 1 In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 8. y = 2y + x2 + 5 In Problems 1-24 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 9. xdydxy=x2sinx In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 10. xdydx+2y=3

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