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Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Differential Equations with Boundary-Value Problems (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 28E 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 Make up a differential equation that does not possess any real solutions.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.49E 50E 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 53E 54E 55E 56E The normal form (5) of an nth-order differential equation is equivalent to (4) whenever both forms have exactly the same solutions. Make up a first-order differential equation for which F(x, y, y) = 0 is not equivalent to the normal form dy/dx = f(x, y).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.Consider the differential equation dy/dx = ex2. (a) Explain why a solution of the DE must be an increasing function on any interval of the x-axis. (b) What are limxdy/dx and limxdy/dx? What does this suggest about a solution curve asx (c) Determine an interval over which a solution curve is concave down and an interval over which the curve is concave up. (d) Sketch the graph of a solution y = (x) of the differential equation whose shape is suggested by parts (a) (c).Consider the differential equation dy/dx = 5 y. (a) Either by inspection or by the method suggested in Problems 3740, find a constant solution 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 the y-axis on which y = (x) is decreasing.61E 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 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. 8. x(/2) = 0, x(/2) = 1 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 20E 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 22E 23E 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)27E 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).]32E (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?34E 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 39E 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 43E 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 45E In Problems 45 and 46 use Problem 55 in Exercises 1.1 and (2) and (3) of this section. Find a function y = f(x) whose second derivative is y = 12x 2 at each point (x, y) on its graph and y = x + 5 is tangent to the graph at the point corresponding to x = 1. (reference problem 51 in exercise 1.1) 46. Find a function whose second derivative is y = 12x 2 at each point (x, y) on its graph and y = x + 5 is tangent to the graph at the point corresponding to x = 1.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 50E 51E 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.9E 10E 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?12E 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?21E 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.25E 26E 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.]30E 31E 32E 33E 34E 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?39E 40E 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)=4RE 5RE 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 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 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.16RE 17RE (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.19RE 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.21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE 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 31RE 32RE 33RE 34RE 35RE 36RE 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 38RE 39RE 40RE 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 5E 6E 7E 8E 9E 10E 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 12E 13E 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 20E 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 22E 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 24E 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)27E 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.32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 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 32E 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 34E (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.39E 40E 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. , y(0) = 1 42E 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.44E 45E 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 48E 49E 50E 51E 52E 53E 54E 55E 56E Suspension Bridge In (16) of Section 1.3 we saw that a mathematical model for the shape of a flexible cable strung between two vertical supports is dydx=WT1,(11) where W denotes the portion of the total vertical load between the points P1 and P2 shown in Figure 1.3.7. The DE (11) is separable under the following conditions that describe a suspension bridge. Let us assume that the x- and y-axes are as shown in Figure 2.2.5that is, the x-axis runs along the horizontal roadbed, and the y-axis passes through (0, a), which is the lowest point on one cable over the span of the bridge, coinciding with the interval [L/2, L/2]. In the case of a suspension bridge, the usual assumption is that the vertical load in (11) is only a uniform roadbed distributed along the horizontal axis. In other words, it is assumed that the weight of all cables is negligible in comparison to the weight of the roadbed and that the weight per unit length of the roadbed (say, pounds per horizontal foot) is a constant . Use this information to set up and solve an appropriate initial-value problem from which the shape (a curve with equation y = (x)) of each of the two cables in a suspension bridge is determined. Express your solution of the IVP in terms of the sag h and span L. See Figure 2.2.5.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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