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Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Differential Equations with Boundary-Value Problems (MindTap Course List)

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. 11. xdydx+4y=x3x 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. 12. (1+x)dydxxy=x+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. 13. x2y + x(x + 2)y = ex 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. 14. xy + (1 + x)y = ex sin 2x 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. 15. y dx 4(x + y6) dy = 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. 16. y dx = (yey 2x) dy 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. 17. cosxdydx+(sinx)y=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. 18. cos2xsinxdydx+(cos3x)y=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. 19. (x+1)dydx+(x+2)y=2xex 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. 20. (x+2)2dydx=58y4xy 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. 21. drd+rsec=cos 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. 22. dPdt+2tP=P+4t2 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. 23. xdydx+(3x+1)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. 24. (x21)dydx+2y=(x+1)2 In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. 25. dydx=x+5y, y(0) = 3 In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. 26. dydx=2x3y,y(0)=13 In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. 27. xy + y = ex, y(1) = 2 In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. 28. ydxdyx=2y2, y(1) = 5 In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. 29. Ldidt+Ri=E, i(0) = i0, L, R, E, i0 constants In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. 30. dTdt=k(TTm), T(0) = T0, k, Tm, T0 constants In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. 31. xdydx+y=4x+1, y(1) = 8 In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. 32. y + 4xy = x3ex, y(0) = 1 In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. 33. x(x+1)dydx+y=lnx, y(1) = 10 In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. 34. x(x+1)dydx+xy=1, y(e) = 1 35E 36E 37E In Problems 3740 proceed as in Example 6 to solve the given initial-value problem. Use a graphing utility to graph the continuous function y(x). 38. dydx+y=f(x), y(0) = 1, where f(x)={1,0x11,x1 39E In Problems 3740 proceed as in Example 6 to solve the given initial-value problem. Use a graphing utility to graph the continuous function y(x). 40. (1+x2)dydx+2xy=f(x), y(0) = 0, where f(x)={x,0x1x,x1 41E In Problems 41 and 42 proceed as in Example 6 to solve the given initial-value problem. Use a graphing utility to graph the continuous function y(x). 42. dydx+P(x)y=0, y(0) = 4, where P(x)={1,0x25,x2 43E In Problems 43 and 44 proceed as in Example 7 and express the solution of the given initial-value problem in terms of erf(x) (Problem 43) and erfc(x) (Problem 44). 44. dydx2xy=1,y(0)=/2 45E 46E 47E The Fresnel sine integral function is defined as S(x)=0xsin(2t2)dt. See Appendix A. Express the solution of the initial-value problem dydx(sinx2)y=0,y(0)=5 in terms of S(x).49E 50E 51E 52E 53E 54E 55E 56E 57E Heart Pacemaker A heart pacemaker consists of a switch, a battery of constant voltage E0, a capacitor with constant capacitance C, and the heart as a resistor with constant resistance R. When the switch is closed, the capacitor charges; when the switch is open, the capacitor discharges, sending an electrical stimulus to the heart. During the time the heart is being stimulated, the voltage E across the heart satisfies the linear differential equation dEdt=1RCE. Solve the DE, subject to E(4) = E0.61E In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 1. (2x 1) dx + (3y + 7) dy = 0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 2. (2x + y) dx (x + 6y) dy = 0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 3. (5x + 4y) dx + (4x 8y3) dy = 0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 4. (sin y y sin x) dx + (cos x + x cos y y) dy = 0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 5. (2xy2 3) dx + (2x2y + 4) dy = 0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 6. (2y1x+cos3x)dydx+yx24x3+3ysin3x=0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 7. (x2 y2) dx + (x2 2xy) dy = 0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 8. (1+lnx+yx)dx=(1lnx)dy In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 9. (x y3 + y2 sin x) dx = (3xy2 + 2y cos x) dy In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 10. (x3 + y3) dx + 3xy2 dy = 0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 11. (ylnyexy)dx+(1y+xlny)dy=0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 12. (3x2y + ey) dx +(x3 + xey 2y) dy = 0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 13. xdydx=2xexy+6x2 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 14. (13y+x)dydx+y=3x1 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 15. (x2y311+9x2)dxdy+x3y2=0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 16. (5y 2x)y 2y = 0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 17. (tan x sin x sin y) dx + cos x cos y dy = 0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 18. (2ysinxcosxy+2y2exy2)dx=(xsin2x4xyexy2)dy In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 19. (4t3y 15t2 y) dt + (t4 + 3y2 t) dy = 0 In Problems 120 determine whether the given differential equation is exact. If it is exact, solve it. 20. (1t+1t2yt2+y2)dt+(yey+tt2+y2)dy=0 In problems 2126 slove the given initial-value problem. 21. (x + y)2 dx + (2xy + x2 1) dy = 0, y(1) = 1 In Problem 2126 solve the given initial-value problem. 22. (ex + y) dx + (2 + x + yey) dy = 0, y(0) = 1 23E In Problems 2126 solve the given initial-value problem. 24. dydx=y21x21,y(2)=2 In Problems 2126 solve the given initial-value problem. 25. (y2 cos x 3x2y 2x) dx + (2y sin x x3 + ln y) dy = 0, y(0) = e 26E In Problems 27 and 28 find the value of k so that the given differential equation is exact. 27. (y3 + kxy4 2x) dx + (3xy2 + 20x2y3) dy = 0 In Problems 27 and 28 find the value of k so that the given differential equation is exact. 28. (6xy3 + cos y) dx + (2kx2y2 x sin y) dy = 0 In Problems 29 and 30 verify that the given differential equation is not exact. Multiply the given differential equation by the indicated integrating factor (x, y) and verify that the new equation is exact. Solve. 29. (xy sin x + 2y cos x) dx + 2x cos x dy = 0; (x, y) = xy In Problems 29 and 30 verify that the given differential equation is not exact. Multiply the given differential equation by the indicated integrating factor (x, y) and verify that the new equation is exact. Solve. 30. (x2 + 2xy y2) dx + (y2 + 2xy x2) dy = 0; (x, y) = (x + y)2 In Problems 3136 solve the given differential equation by finding, as in Example 4, an appropriate integrating factor. 31. (2y2 + 3x) dx + 2xy dy = 0 In Problems 3136 solve the given differential equation by finding, as in Example 4, an appropriate integrating factor. 32. y(x + y + 1) dx + (x + 2y) dy = 0 In Problems 3136 solve the given differential equation by finding, as in Example 4, an appropriate integrating factor. 33. 6xy dx + (4y + 9x2) dy = 0 In Problems 3136 solve the given differential equation by finding, as in Example 4, an appropriate integrating factor. 34. cosxdx+(1+2y)sinxdy=0 In Problems 3136 solve the given differential equation by finding, as in Example 4, an appropriate integrating factor. 35. (10 6y + e 3x) dx 2 dy = 0 In Problems 3136 solve the given differential equation by finding, as in Example 4, an appropriate integrating factor. 36. (y2 + xy3) dx + (5y2 xy + y3 sin y) dy = 0 In Problems 37 and 38 solve the given initial-value problem by finding, as in Example 4, an appropriate integrating factor. 37. x dx + (x2y + 4y) dy = 0, y(4) = 0 In Problems 37 and 38 solve the given initial-value problem by finding, as in Example 4, an appropriate integrating factor. 38. (x2 + y2 5) dx = (y + xy) dy, y(0) = 1 (a) Show that a one-parameter family of solutions of the equation (4xy + 3x2) dx + (2y + 2x2) dy = 0 is x3 + 2x2y + y2 = c. (b) Show that the initial conditions y(0) = 2 and y(1) = 1 determine the same implicit solution. (c) Find explicit solutions y1(x) and y2(x) of the differential equation in part (a) such that y1(0) = 2 and y2(1) = 1. Use a graphing utility to graph y1(x) and y2(x).40E 41E 42E Differential equations are sometimes solved by having a clever idea. Here is a little exercise in cleverness: Although the differential equation (xx2+y2)dx+ydy=0 is not exact, show how the rearrangement (x dx + y dy)/x2+y2 = dx and the observation 12d(x2 + y2) = x dx + y dy can lead to a solution.44E Falling Chain A portion of a uniform chain of length 8 ft is loosely coiled around a peg at the edge of a high horizontal platform, and the remaining portion of the chain hangs at rest over the edge of the platform. See Figure 2.4.2. Suppose that the length of the overhanging chain is 3 ft, that the chain weighs 2 lb/ft, and that the positive direction is downward. Starting at t = 0 seconds, the weight of the overhanging portion causes the chain on the table to uncoil smoothly and to fall to the floor. If x(t) denotes the length of the chain overhanging the table at time t 0, then v = dx/dt is its velocity. When all resistive forces are ignored, it can be shown that a mathematical model relating v to x is given by xvdvdx+v2=32x. (a) Rewrite this model in differential form. Proceed as in Problems 3136 and solve the DE for v in terms of x by finding an appropriate integrating factor. Find an explicit solution v(x). (b) Determine the velocity with which the chain leaves the platform. FIGURE 2.4.2 Uncoiling chain in Problem 45 Each DE in Problems 114 is homogeneous. In Problems 110 solve the given differential equation by using an appropriate substitution. 1. (x y) dx + x dy = 0 In Problems 110 solve the given differential equation by using an appropriate substitution. 2. (x + y) dx + x dy = 0 In Problems 110 solve the given differential equation by using an appropriate substitution. 3. x dx + (y 2x) dy = 0 In Problems 1-10 solve the given differential equation by using an appropriate substitution. 4. y dx = 2(x + y)dy In Problems 110 solve the given differential equation by using an appropriate substitution. 5. (y2 + yx) dx x2 dy = 0 In Problems 1-10 solve the given differential equation by using an appropriate substitution. 6. (y2 + yx) dx + x2 dy = 0 In Problems 110 solve the given differential equation by using an appropriate substitution. 7. dydx=yxy+x In Problems 110 solve the given differential equation by using an appropriate substitution. 8. dydx=x+3y3x+y In Problems 110 solve the given differential equation by using an appropriate substitution. 9. ydx+(x+xy)dy=0 In Problems 110 solve the given differential equation by using an appropriate substitution. 10. xdydx=yx2y2, x 0 In Problems 1114 solve the given initial-value problem. 11. xy2dydx=y3x3, y(1) = 2 In Problems 1114 solve the given initial-value problem. 12. (x2+2y2)dxdy=xy, y(1) = 1 In Problems 1114 solve the given initial-value problem. 13. (x + yey/x) dx xey/x dy = 0, y(1) = 0 In Problems 1114 solve the given initial-value problem. 14. y dx + x(ln x ln y 1) dy = 0, y(1) = e In Problems 1520 solve the given differential equation by using an appropriate substitution. 15. xdydx+y=1y2 In Problems 1520 solve the given differential equation by using an appropriate substitution. 16. dydxy=exy2 In Problems 1520 solve the given differential equation by using an appropriate substitution. 17. dydx=y(xy31)In Problems 1520 solve the given differential equation by using an appropriate substitution. 18. xdydx(1+x)y=xy2 19E 20E In Problems 21 and 22 solve the given initial-value problem. 21. x2dydx2xy=3y4,y(1)=12 In Problems 21 and 22 solve the given initial-value problem. 22. y1/2dydx+y3/2=1, y(0) = 4 In Problems 2328 solve the given differential equation by using an appropriate substitution. 23. dydx=(x+y+1)2 In Problems 2328 solve the given differential equation by using an appropriate substitution. 24. dydx=1xyx+y In Problems 2328 solve the given differential equation by using an appropriate substitution. 25. dydx=tan2(x+y)In Problems 2328 solve the given differential equation by using an appropriate substitution. 26. dydx=sin(x+y)In Problems 2328 solve the given differential equation by using an appropriate substitution. 27. dydx=2+y2x+3 In Problems 2328 solve the given differential equation by using an appropriate substitution. 28. dydx=1+eyx+5 dydx=cos(x+y), y(0) = /4 In Problems 29 and 30 solve the given initial-value problem. 30. dydx=3x+2y3x+2y+2,y(1)=1 Explain why it is always possible to express any homogeneous differential equation M(x, y) dx + N(x, y) dy = 0 in the form dydx=F(yx). You might start by proving that M(x, y) = xM(1, y/x) and N(x, y) = xN(1, y/x).Put the homogeneous differential equation (5x22y2)dxxydy=0 into the form given in Problem 31.(a) Determine two singular solutions of the DE in Problem 10. (b) If the initial condition y(5) = 0 is as prescribed in Problem 10, then what is the largest interval I over which the solution is defined? Use a graphing utility to graph the solution curve for the IVP.34E The differential equation dy/dx = P(x) + Q(x)y + R(x)y2 is known as Riccatis equation. (a) A Riccati equation can be solved by a succession of two substitutions provided that we know a particular solution y1 of the equation. Show that the substitution y = y1 + u reduces Riccatis equation to a Bernoulli equation (4) with n = 2. The Bernoulli equation can then be reduced to a linear equation by the substitution w = u1. (b) Find a one-parameter family of solutions for the differential equation dydx=4x21xy+y2 where y1 = 2/x is a known solution of the equation.36E Falling Chain In Problem 45 in Exercises 2.4 we saw that a mathematical model for the velocity v of a chain slipping off the edge of a high horizontal platform is xvdvdx+v2=32x. In that problem you were asked to solve the DE by converting it into an exact equation using an integrating factor. This time solve the DE using the fact that it is a Bernoulli equation.Population Growth In the study of population dynamics one of the most famous models for a growing but bounded population is the logistic equation dPdt=P(abP), where a and b are positive constants. Although we will come back to this equation and solve it by an alternative method in Section 3.2, solve the DE this first time using the fact that it is a Bernoulli equation.In Problems 1 and 2 use Eulers method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of (3) by hand, first using h = 0.1 and then using h = 0.05. 1. y = 2x 3y + 1, y(1) = 5; y(1.2)In Problems 1 and 2 use Eulers method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of (3) by hand, first using h = 0.1 and then using h = 0.05. 2. y = x + y2, y(0) = 0; y(0.2)In Problems 3 and 4 use Eulers method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05. Find an explicit solution for each initial-value problem and then construct tables similar to Tables 2.6.3 and 2.6.4. y = 2xy, y(1) = 1; y(1.5)In Problems 3 and 4 use Eulers method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05. Find an explicit solution for each initial-value problem and then construct tables similar to Tables 2.6.3 and 2.6.4. y = 2xy, y(1) = 1; y(1.5)In Problems 510 use a numerical solver and Eulers method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05. 5. y=ey,y(0)=0;y(0.5)6E In Problems 510 use a numerical solver and Eulers method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05. 7. y = (x y)2, y(0) = 0.5; y(0.5)8E In Problems 510 use a numerical solver and Eulers method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05. 9. y=xy2yx,y(1)=1;y(1.5)In Problems 510 use a numerical solver and Eulers method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05. 10. y=yy2,y(0)=0.5;y(0.5)Answer Problems 112 without referring back to the text. Fill in the blanks or answer true or false. 1. The linear DE, y ky = A, where k and A are constants, is autonomous. The critical point __________ of the equation is a(n) __________ (attractor or repeller) for k 0 and a(n) __________ (attractor or repeller) for k 0.2RE 3RE 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE In Problems 13 and 14 construct an autonomous first-order differential equation dy/dx = f(y) whose phase portrait is consistent with the given figure. FIGURE 2.R.1 Graph for Problem 13 13.In Problems 13 and 14 construct an autonomous first-order differential equation dy/dx = f(y) whose phase portrait is consistent with the given figure. 14. FIGURE 2.R.2 Graph for Problem 14 15RE 16RE 17RE Classify each differential equation as separable, exact, linear, homogeneous, or Bernoulli. Some equations may be more the one kind. Do not solve. (a) dydx=xyx (b) dydx=1yx (c) (x+1)dydx=y+10 (d) dydx=1x(xy) (e) dydx=y2+yx2+x (f) dydx=5y+y2 (g) y dx = (y xy2) dy (h) xdydx=yex/yx (i) xy y + y2 = 2x (j) 2xy y + y2 = 2x2 (k) y dx + x dy = 0 (l) (x2+2yx)dx=(3lnx2)dy (m) dydx=xy+yx+1 (n) yx2dydx+e2x3+y2=0 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE 30RE 31RE 32RE In Problems 33 and 34 solve the given initial-value problem and give the largest interval I on which the solution is defined. 33. sinxdydx+(cosx)y=0, y(7/6) = 2 In Problems 33 and 34 solve the given initial-value problem and give the largest interval I on which the solution is defined. 34. dydt+2(t+1)y2=0,y(0)=18 35RE 36RE 37RE 38RE The population of a community is known to increase at a rate proportional to the number of people present at time t. If an initial population P0 has doubled in 5 years, how long will it take to triple? To quadruple?Suppose it is known that the population of the community in Problem 1 is 10,000 after 3 years. What was the initial population P0? What will be the population in 10 years? How fast is the population growing at t = 10? 1. The population of a community is known to increase at a rate proportional to the number of people present at time t. If an initial population P0 has doubled in 5 years, how long will it take to triple? To quadruple?The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 15% in 10 years. What will be the population in 30 years? How fast is the population growing at t = 30?The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. After 3 hours it is observed that 400 bacteria are present. After 10 hours 2000 bacteria are present. What was the initial number of bacteria?The radioactive isotope of lead, Pb-209, decays at a rate proportional to the amount present at time t and has a half-life of 3.3 hours. If 1 gram of this isotope is present initially, how long will it take for 90% of the lead to decay?Initially 100 milligrams of a radioactive substance was present. After 6 hours the mass had decreased by 3%. If the rate of decay is proportional to the amount of the substance present at time t, find the amount remaining after 24 hours.Determine the half-life of the radioactive substance described in Problem 6. 6. Initially 100 milligrams of a radioactive substance was present. After 6 hours the mass had decreased by 3%. If the rate of decay is proportional to the amount of the substance present at time t, find the amount remaining after 24 hours.(a) Consider the initial-value problem dA/dt = kA, A(0) = A0 as the model for the decay of a radioactive substance. Show that, in general, the half-life T of the substance is T = (ln 2)/k. (b) Show that the solution of the initial-value problem in part (a) can be written A(t) = A02t/T. (c) If a radioactive substance has the half-life T given in part (a), how long will it take an initial amount A0 of the substance to decay to 18A0?When a vertical beam of light passes through a transparent medium, the rate at which its intensity I decreases is proportional to I(t), where t represents the thickness of the medium (in feet). In clear seawater, the intensity 3 feet below the surface is 25% of the initial intensity I0 of the incident beam. What is the intensity of the beam 15 feet below the surface?10E Carbon Dating Archaeologists used pieces of burned wood, or charcoal, found at the site to date prehistoric paintings and drawings on walls and ceilings of a cave in Lascaux, France. See Figure 3.1.11. Use the information on page 87 to determine the approximate age of a piece of burned wood, if it was found that 85.5% of the C-14 found in living trees of the same type had decayed. FIGURE 3.1.11 Cave wall painting in Problem 11 Rock painting showing a horse and a cow, c. 17000 BC (cave painting), Prehistoric/Caves of Lascaux, Dordogne, France/Bridgeman Images The Shroud of Turin, which shows the negative image of the body of a man who appears to have been crucified, is believed by many to be the burial shroud of Jesus of Nazareth. See Figure 3.1.12. In 1988 the Vatican granted permission to have the shroud carbon-dated. Three independent scientific laboratories analyzed the cloth and concluded that the shroud was approximately 660 years old, an age consistent with its historical appearance. Using this age, determine what percentage of the original amount of C-14 remained in the cloth as of 1988. FIGURE 3.1.12 Shroud image in Problem 12 Source: Wikipedia.org Newtons Law of Cooling/Warming A thermometer is removed from a room where the temperature is 70 F and is taken outside, where the air temperature is 10 F. After one-half minute the thermometer reads 50 F. What is the reading of the thermometer at t = 1 min? How long will it take for the thermometer to reach 15 F?A thermometer is taken from an inside room to the outside, where the air temperature is 5 F. After 1 minute the thermometer reads 55 F, and after 5 minutes it reads 30 F. What is the initial temperature of the inside room?A small metal bar, whose initial temperature was 20 C, is dropped into a large container of boiling water. How long will it take the bar to reach 90 C if it is known that its temperature increases 2 in 1 second? How long will it take the bar to reach 98 C?Two large containers A and B of the same size are filled with different fluids. The fluids in containers A and B are maintained at 0 C and 100 C, respectively. A small metal bar, whose initial temperature is 100 C, is lowered into container A. After 1 minute the temperature of the bar is 90 C. After 2 minutes the bar is removed and instantly transferred to the other container. After 1 minute in container B the temperature of the bar rises 10. How long, measured from the start of the entire process, will it take the bar to reach 99.9 C?A thermometer reading 70 F is placed in an oven preheated to a constant temperature. Through a glass window in the oven door, an observer records that the thermometer reads 110 F after 12 minute and 145 F after 1 minute. How hot is the oven?At t = 0 a sealed test tube containing a chemical is immersed in a liquid bath. The initial temperature of the chemical in the test tube is 80 F. The liquid bath has a controlled temperature (measured in degrees Fahrenheit) given by Tm(t) = 100 40e0.1t, t 0, where t is measured in minutes. (a) Assume that k = 0.1 in (2). Before solving the IVP, describe in words what you expect the temperature T(t) of the chemical to be like in the short term. In the long term. (b) Solve the initial-value problem. Use a graphing utility to plot the graph of T(t) on time intervals of various lengths. Do the graphs agree with your predictions in part (a)?A dead body was found within a closed room of a house where the temperature was a constant 70 F. At the time of discovery the core temperature of the body was determined to be 85 F. One hour later a second measurement showed that the core temperature of the body was 80 F. Assume that the time of death corresponds to t = 0 and that the core temperature at that time was 98.6 F. Determine how many hours elapsed before the body was found. [Hint: Let t1 0 denote the time that the body was discovered.]The rate at which a body cools also depends on its exposed surface area S. If S is a constant, then a modification of (2) is dTdt=kS(TTm), where k 0 and Tm is a constant. Suppose that two cups A and B are filled with coffee at the same time. Initially, the temperature of the coffee is 150 F. The exposed surface area of the coffee in cup B is twice the surface area of the coffee in cup A. After 30 min the temperature of the coffee in cup A is 100 F. If Tm = 70 F, then what is the temperature of the coffee in cup B after 30 min?A tank contains 200 liters of fluid in which 30 grams of salt is dissolved. Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of 4 L/min; the well-mixed solution is pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t.Solve Problem 21 assuming that pure water is pumped into the tank. 21. A tank contains 200 liters of fluid in which 30 grams of salt is dissolved. Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of 4 L/min; the well-mixed solution is pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t.A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well-mixed solution is pumped out at the same rate. Find the number A(t) of pounds of salt in the tank at time t.In Problem 23, what is the concentration c(t) of the salt in the tank at time t? At t = 5 min? What is the concentration of the salt in the tank after a long time, that is, as t ? At what time is the concentration of the salt in the tank equal to one-half this limiting value? 23. A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well-mixed solution is pumped out at the same rate. Find the number A(t) of pounds of salt in the tank at time t.Solve Problem 23 under the assumption that the solution is pumped out at a faster rate of 10 gal/min. When is the tank empty? 23. A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well-mixed solution is pumped out at the same rate. Find the number A(t) of pounds of salt in the tank at time t.Determine the amount of salt in the tank at time t in Example 5 if the concentration of salt in the inflow is variable and given by cin(t) = 2 + sin(t/4) lb/gal. Without actually graphing, conjecture what the solution curve of the IVP should look like. Then use a graphing utility to plot the graph of the solution on the interval [0, 300]. Repeat for the interval [0, 600] and compare your graph with that in Figure 3.1.6(a).A large tank is partially filled with 100 gallons of fluid in which 10 pounds of salt is dissolved. Brine containing 12 pound of salt per gallon is pumped into the tank at a rate of 6 gal/min. The well-mixed solution is then pumped out at a slower rate of 4 gal/min. Find the number of pounds of salt in the tank after 30 minutes.In Example 5 the size of the tank containing the salt mixture was not given. Suppose, as in the discussion following Example 5, that the rate at which brine is pumped into the tank is 3 gal/min but that the well-stirred solution is pumped out at a rate of 2 gal/min. It stands to reason that since brine is accumulating in the tank at the rate of 1 gal/min, any finite tank must eventually overflow. Now suppose that the tank has an open top and has a total capacity of 400 gallons. (a) When will the tank overflow? (b) What will be the number of pounds of salt in the tank at the instant it overflows? (c) Assume that although the tank is overflowing, brine solution continues to be pumped in at a rate of 3 gal/min and the well-stirred solution continues to be pumped out at a rate of 2 gal/min. Devise a method for determining the number of pounds of salt in the tank at t =150 minutes. (d) Determine the number of pounds of salt in the tank as t Does your answer agree with your intuition? (e) Use a graphing utility to plot the graph of A(t) on the interval [0, 500).A 30-volt electromotive force is applied to an LR-series circuit in which the inductance is 0.1 henry and the resistance is 50 ohms. Find the current i(t) if i(0) = 0. Determine the current as t .Solve equation (7) under the assumption that E(t) = E0 sin t and i(0) = i0.A 100-volt electromotive force is applied to an RC-series circuit in which the resistance is 200 ohms and the capacitance is 104 farad. Find the charge q(t) on the capacitor if q(0) = 0. Find the current i(t).A 200-volt electromotive force is applied to an RC-series circuit in which the resistance is 1000 ohms and the capacitance is 5 106 farad. Find the charge q(t) on the capacitor if i(0) = 0.4. Determine the charge and current at t = 0.005 s. Determine the charge as t .An electromotive force E(t)={120,0t200,t20 is applied to an LR-series circuit in which the inductance is 20 henries and the resistance is 2 ohms. Find the current i(t) if i(0) = 0.An LR-series circuit has a variable inductor with the inductance defined by L(t)={1110t,0t100,t10. Find the current i(t) if the resistance is 0.2 ohm, the impressed voltage is E(t) = 4, and i(0) = 0. Graph i(t).Air Resistance In (14) of Section 1.3 we saw that a differential equation describing the velocity v of a falling mass subject to air resistance proportional to the instantaneous velocity is mdvdt=mgkv, where k 0 is a constant of proportionality. The positive direction is downward. (a) Solve the equation subject to the initial condition v(0) = v0. (b) Use the solution in pan (a) to determine the limiting, or terminal, velocity of the mass. We saw how to determine the terminal velocity without solving the DE in Problem 40 in Exercises 2.1. (c) If the distance s, measured from the point where the mass was released above ground. is related to velocity v by ds/dt = v(t), find an explicit expression for s(t) if s(0) = 0.How High?No Air Resistance Suppose a small cannonball weighing 16 pounds is shot vertically upward, as shown in Figure 3.1.13, with an initial velocity v0 = 300 ft/s. The answer to the question How high does the cannonball go? depends on whether we take air resistance into account. (a) Suppose air resistance is ignored. If the positive direction is upward, then a model for the state of the cannonball is given by d2s/dt2 = g (equation (12) of Section 1.3). Since ds/dt = v(t) the last differential equation is the same as dv/dt = g, where we take g = 32 ft/s2. Find the velocity v(t) of the cannonball at time t. (b) Use the result obtained in part (a) to determine the height s(t) of the cannonball measured from ground level. Find the maximum height attained by the cannonball. FIGURE 3.1.13 Find the maximum height of the cannonball in Problem 36 How High?Linear Air Resistance Repeat Problem 36, but this time assume that air resistance is proportional to instantaneous velocity. It stands to reason that the maximum height attained by the cannonball must be less than that in part (b) of Problem 36. Show this by supposing that the constant of proportionality is k = 0.0025. [Hint: Slightly modify the differential equation in Problem 35.]Skydiving A skydiver weighs 125 pounds, and her parachute and equipment combined weigh another 35 pounds. After exiting from a plane at an altitude of 15,000 feet, she waits 15 seconds and opens her parachute. Assume that the constant of proportionality in the model in Problem 35 has the value k = 0.5 during free fall and k = 10 after the parachute is opened. Assume that her initial velocity on leaving the plane is zero. What is her velocity and how far has she traveled 20 seconds after leaving the plane? See Figure 3.1.14. How does her velocity at 20 seconds compare with her terminal velocity? How long does it take her to reach the ground? [Hint: Think in terms of two distinct IVPs.) FIGURE 3.1.14 Find the time to reach the ground in Problem 38 39E Rocket MotionContinued In Problem 39 suppose of the rockets initial mass m0 that 50 kg is the mass of the fuel. (a) What is the burnout time tb, or the time at which all the fuel is consumed? (b) What is the velocity of the rocket at burnout? (c) What is the height of the rocket at burnout? (d) Why would you expect the rocket to attain an altitude higher than the number in part (b)? (e) After burnout what is a mathematical model for the velocity of the rocket? 39. Rocket Motion Suppose a small single-stage rocket of total mass m(t) is launched vertically, the positive direction is upward, the air resistance is linear, and the rocket consumes its fuel at a constant rate. In Problem 22 of Exercises 1.3 you were asked to use Newtons second law of motion in the form given in (17) of that exercise set to show that a mathematical model for the velocity v(t) of the rocket is given by dvdt+km0tv=g+Rm0t, where k is the air resistance constant of proportionality, is the constant rate at which fuel is consumed, R is the thrust of the rocket, m(t) = m0 t, m0 is the total mass of the rocket at t = 0, and g is the acceleration due to gravity. (a) Find the velocity v(t) of the rocket if m0 = 200 kg, R = 2000 N, = 1 kg/s, g = 9.8 m/s2, k = 3 kg/s, and v(0) = 0. (b) Use ds/dt = v and the result in part (a) to find the height s(t) of the rocket at time t.Evaporating Raindrop As a raindrop falls, it evaporates while retaining its spherical shape. If we make the further assumptions that the rate at which the raindrop evaporates is proportional to its surface area and that air resistance is negligible, then a model for the velocity v(t) of the raindrop is dvdt+3(k/)(k/)t+r0v=g. Here is the density of water, r0 is the radius of the raindrop at t = 0, k 0 is the constant of proportionality, and the downward direction is taken to be the positive direction. (a) Solve for v(t) if the raindrop falls from rest. (b) Reread Problem 36 of Exercises 1.3 and then show that the radius of the raindrop at time t is r(t) = (k/)t + r0. (c) If r0 = 0.01 ft and r = 0.007 ft 10 seconds after the raindrop falls from a cloud, determine the time at which the raindrop has evaporated completely.42E 43E Constant-Harvest model A model that describes the population of a fishery in which harvesting takes place at a constant rate is given by dPdt=kPh, where k and h are positive constants. (a) Solve the DE subject to P(0) = P0. (b) Describe the behavior of the population P(t) for increasing time in the three cases P0 h/k, P0 = h/k, and 0 P0 h/k. (c) Use the results from part (b) to determine whether the fish population will ever go extinct in finite time, that is, whether there exists a time T 0 such that P(T) = 0. If the population goes extinct, then find T.Drug Dissemination A mathematical model for the rate at which a drug disseminates into the bloodstream is given by dxdt=rkx, where r and k are positive constants. The function x(t) describes the concentration of the drug in the bloodstream at time t. (a) Since the DE is autonomous, use the phase portrait concept of Section 2.1 to find the limiting value of x(t) as t . (b) Solve the DE subject to x(0) = 0. Sketch the graph of x(t) and verify your prediction in part (a). At what time is the concentration one-half this limiting value?46E Heart Pacemaker A heart pacemaker, shown in Figure 3.1.15, consists of a switch, a battery, a capacitor, and the heart as a resistor. When the switch S is at P, the capacitor charges; when S is at Q, the capacitor discharges, sending an electrical stimulus to the heart. In Problem 58 in Exercises 2.3 we saw that during this time the electrical stimulus is being applied to the heart, the voltage E across the heart satisfies the linear DE dEdt=1RCE. (a) Let us assume that over the time interval of length t1, 0 t t1, the switch S is at position P shown in Figure 3.1.15 and the capacitor is being charged. When the switch is moved to position Q at time t1 the capacitor discharges, sending an impulse to the heart over the time interval of length t2: t1 t t1 + t2. Thus over the initial charging/ discharging interval 0 t t1 + t2 the voltage to the heart is actually modeled by the piecewise-linear differential equation dEdt={0,0tt11RCE,t1tt1+t2. By moving S between P and Q, the charging and discharging over time intervals of lengths t1 and t2 is FIGURE 3.1.15 Model of a pacemaker in Problem 47 repeated indefinitely. Suppose t1 = 4 s, t2 = 2 s, E0 = 12 V, and E(0) = 0, E(4) = 12, E(6) = 0, E(10) = 12, E(12) = 0, and so on. Solve for E(t) for 0 t 24. (b) Suppose for the sake of illustration that R = C = 1. Use a graphing utility to graph the solution for the IVP in part (a) for 0 t 24.Sliding Box (a) A box of mass m slides down an inclined plane that makes an angle with the horizontal as shown in Figure 3.1.16. Find a differential equation for the velocity v(t) of the box at time t in each of the following three cases: (i) No sliding friction and no air resistance (ii) With sliding friction and no air resistance (iii) With sliding friction and air resistance In cases (ii) and (iii), use the fact that the force of friction opposing the motion of the box is N, where is the coefficient of sliding friction and N is the normal component of the weight of the box. In case (iii) assume that air resistance is proportional to the instantaneous velocity. (b) In part (a), suppose that the box weighs 96 pounds, that the angle of inclination of the plane is = 30, that the coefficient of sliding friction is =3/4, and that the additional retarding force due to air resistance is numerically equal to 14v. Solve the differential equation in each of the three cases, assuming that the box starts from rest from the highest point 50 ft above ground. FIGURE 3.1.16 Box sliding down inclined plane in Problem 48 Sliding BoxContinued (a) In Problem 48 let s(t) be the distance measured down the inclined plane from the highest point. Use ds/dt = v(t) and the solution for each of the three cases in part (b) of Problem 48 to find the time that it takes the box to slide completely down the inclined plane. A root-finding application of a CAS may be useful here. (b) In the case in which there is friction ( 0) but no air resistance, explain why the box will not slide down the plane starting from rest from the highest point above ground when the inclination angle satisfies tan . (c) The box will slide downward on the plane when tan if it is given an initial velocity v(0) = v0 0. Suppose that =3/4 and, = 23. Verify that tan How far will the box slide down the plane if v0 = 1 ft/s? (d) Using the values =3/4 and = 23, approximate the smallest initial velocity v0 that can be given to the box so that, starting at the highest point 50 ft above ground, it will slide completely down the inclined plane. Then find the corresponding time it takes to slide down the plane.50E The number N(t) of supermarkets throughout the country that are using a computerized checkout system is described by the initial-value problem dNdt=N(10.0005N),N(0)=1. (a) Use the phase portrait concept of Section 2.1 to predict how many supermarkets are expected to adopt the new procedure over a long period of time. By hand, sketch a solution curve of the given initial-value problem. (b) Solve the initial-value problem and then use a graphing utility to verify the solution curve in part (a). How many companies are expected to adopt the new technology when t = 10?The number N(t) of people in a community who are exposed to a particular advertisement is governed by the logistic equation. Initially, N(0) = 500, and it is observed that N(1) = 1000. Solve for N(t) if it is predicted that the limiting number of people in the community who will see the advertisement is 50,000.3E (a) Census data for the United States between 1790 and 1950 are given in Table 3.2.1. Construct a logistic population model using the data from 1790, 1850, and 1910. (b) Construct a table comparing actual census population with the population predicted by the model in part (a). Compute the error and the percentage error for each entry pair. TABLE 3.2.1 (a) If a constant number h of fish are harvested from a fishery per unit time, then a model for the population P(t) of the fishery at time t is given by dPdt=P(abP)h,P(0)=P0, where a, b, h, and P0 are positive constants. Suppose a = 5, b = 1, and h = 4. Since the DE is autonomous, use the phase portrait concept of Section 2.1 to sketch representative solution curves corresponding to the cases P0 4, 1 P0 4, and 0 P0 1. Determine the long-term behavior of the population in each case. (b) Solve the IVP in part (a). Verify the results of your phase portrait in part (a) by using a graphing utility to plot the graph of P(t) with an initial condition taken from each of the three intervals given. (c) Use the information in parts (a) and (b) to determine whether the fishery population becomes extinct in finite time. If so, find that time.Investigate the harvesting model in Problem 5 both qualitatively and analytically in the case a = 5, b = 1, h=254. Determine whether the population becomes extinct in finite time. If so, find that time.Repeat Problem 6 in the case a = 5, b = 1, h = 7.8E Two chemicals A and B are combined to form a chemical C. The rate, or velocity, of the reaction is proportional to the product of the instantaneous amounts of A and B not converted to chemical C. Initially, there are 40 grams of A and 50 grams of B, and for each gram of B, 2 grams of A is used. It is observed that 10 grams of C is formed in 5 minutes. How much is formed in 20 minutes? What is the limiting amount of C after a long time? How much of chemicals A and B remains after a long time?Solve Problem 9 if 100 grams of chemical A is present initially. At what time is chemical C half-formed?Leaking cylindrical tank A tank in the form of a right-circular cylinder standing on end is leaking water through a circular hole in its bottom. As we saw in (10) of Section 1.3, when friction and contraction of water at the hole are ignored, the height h of water in the tank is described by dhdt=AhAw2gh, where Aw and Ah are the cross-sectional areas of the water and the hole, respectively. (a) Solve the DE if the initial height of the water is H. By hand, sketch the graph of h(t) and give its interval I of definition in terms of the symbols Aw, Ah, and H. Use g = 32 ft/s2. (b) Suppose the tank is 10 feet high and has radius 2 feet and the circular hole has radius 12 inch. If the tank is initially full, how long will it take to empty?Leaking cylindrical tankcontinued When friction and contraction of the water at the hole are taken into account, the model in Problem 11 becomes dhdt=cAhAw2gh, where 0 c 1. How long will it take the tank in Problem 11(b) to empty if c = 0.6? See Problem 13 in Exercises 1.3. 11. Leaking cylindrical tank A tank in the form of a right-circular cylinder standing on end is leaking water through a circular hole in its bottom. As we saw in (10) of Section 1.3, when friction and contraction of water at the hole are ignored, the height h of water in the tank is described by dhdt=AhAw2gh, where Aw and Ah are the cross-sectional areas of the water and the hole, respectively. (b) Suppose the tank is 10 feet high and has radius 2 feet and the circular hole has radius 12 inch. If the tank is initially full, how long will it take to empty?Leaking Conical Tank A tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom. (a) Suppose the tank is 20 feet high and has radius 8 feet and the circular hole has radius 2 inches. In Problem 14 in Exercises 1.3 you were asked to show that the differential equation governing the height h of water leaking from a tank is dhdt=56h3/2. In this model, friction and contraction of the water at the hole were taken into account with c = 0.6, and g was taken to be 32 ft/s2. See Figure 1.3.12. If the tank is initially full, how long will it take the tank to empty? (b) Suppose the tank has a verte x angle of 60 and the circular hole has radius 2 inches. Determine the differential equation governing the height h of water. Use c = 0.6 and g = 32 ft/s2. If the height of the water is initially 9 feet, how long will it take the tank to empty?Inverted Conical Tank Suppose that the conical tank in Problem 13(a) is inverted, as shown in Figure 3.2.5, and that water leaks out a circular hole of radius 2 inches in the center of its circular base. Is the time it takes to empty a full tank the same as for the tank with vertex down in Problem 13? Take the friction/contraction coefficient to be c = 0.6 and g = 32 ft/s2. FIGURE 3.2.5 Inverted conical tank in Problem 14 Air Resistance A differential equation for the velocity v of a falling mass m subjected to air resistance proportional to the square of the instantaneous velocity is mdvdt=mgkv2, where k 0 is a constant of proportionality. The positive direction is downward. (a) Solve the equation subject to the initial condition v(0) = v0. (b) Use the solution in part (a) to determine the limiting, or terminal, velocity of the mass. We saw how to determine the terminal velocity without solving the DE in Problem 41 in Exercises 2.1. (c) If the distance s, measured from the point where the mass was released above ground, is related to velocity v by ds/dt = v(t), find an explicit expression for s(t) if s(0) = 0.How High?Nonlinear Air Resistance Consider the 16-pound cannonball shot vertically upward in Problems 36 and 37 in Exercises 3.1 with an initial velocity v0 = 300 ft/s. Determine the maximum height attained by the cannonball if air resistance is assumed to be proportional to the square of the instantaneous velocity. Assume that the positive direction is upward and take k = 0.0003. [Hint: Slightly modify the DE in Problem 15.]17E 18E 19E Evaporation An outdoor decorative pond in the shape of a hemispherical tank is to be filled with water pumped into the tank through an inlet in its bottom. Suppose that the radius of the tank is R = 10 ft, that water is pumped in at a rate of ft3/min, and that the tank is initially empty. See Figure 3.2.6. As the tank fills, it loses water through evaporation. Assume that the rate of evaporation is proportional to the area A of the surface of the water and that the constant of proportionality is k = 0.01. (a) The rate of change dV/dt of the volume of the water at time t is a net rate. Use this net rate to determine a differential equation for the height h of the water at time t. The volume of the water shown in the figure is V=Rh213h3, where R = 10. Express the area of the surface of the water A = r2 in terms of h. (b) Solve the differential equation in part (a). Graph the solution. (c) If there were no evaporation, how long would it take the tank to fill? (d) With evaporation, what is the depth of the water at the time found in part (c)? Will the tank ever be filled? Prove your assertion. Figure 3.2.6 Decorative pond in Problem 20 Doomsday equation Consider the differential equation dPdt=kP1+c, where k 0 and c 0. In Section 3.1 we saw that in the case c = 0 the linear differential equation dP/dt = kP is a mathematical model of a population P(t) that exhibits unbounded growth over the infinite time interval [0, ), that is, P(t) as t . See Example 1 on page 85. (a) Suppose for c = 0.01 that the nonlinear differential equation dPdt=kP1.01,k0, is a mathematical model for a population of small animals, where time t is measured in months. Solve the differential equation subject to the initial condition P(0) = 10 and the fact that the animal population has doubled in 5 months. (b) The differential equation in part (a) is called a doomsday equation because the population P(t) exhibits unbounded growth over a finite time interval (0, T), that is, there is some time T such that P(t) as t T. Find T. (c) From part (a), what is P(50)? P(100)?Doomsday or extinction Suppose the population model (4) is modified to be dPdt=P(bPa). (a) If a 0, b 0 show by means of a phase portrait (see page 40) that, depending on the initial condition P(0) = P0, the mathematical model could include a doomsday scenario (P(t) ) or an extinction scenario (P(t) 0). (b) Solve the initial-value problem dPdt=P(0.0005P0.1),P(0)=300. Show that this model predicts a doomsday for the population in a finite time T. (c) Solve the differential equation in part (b) subject to the initial condition P(0) = 100. Show that this model predicts extinction for the population as t . dPdt=P(abP).26E 27E 28E 29E 30E 31E 32E 33E 34E 35E We have not discussed methods by which systems of first-order differential equations can be solved. Nevertheless, systems such as (2) can be solved with no knowledge other than how to solve a single linear first-order equation. Find a solution of (2) subject to the initial conditions x(0) = x0, y(0) = 0, z(0) = 0.2E 3E Construct a mathematical model for a radioactive series of four elements W, X, Y, and Z, where Z is a stable element. where 1 and 2 are positive constants of proportionality. By proceeding as in Problem 1 we can solve the foregoing mathematical model.5E 6E Consider two tanks A and B, with liquid being pumped in and out at the same rates, as described by the system of equations (3). What is the system of differential equations if, instead of pure water, a brine solution containing 2 pounds of salt per gallon is pumped into tank A?Use the information given in Figure 3.3.6 to construct a mathematical model for the number of pounds of salt x1(t), x2(t), and x3(t) at time t in tanks A, B, and C, respectively. FIGURE 3.3.6 Mixing tanks in Problem 8 Two very large tanks A and B are each partially filled with 100 gallons of brine. Initially, 100 pounds of salt is dissolved in the solution in tank A and 50 pounds of salt is dissolved in the solution in tank B. The system is closed in that the well-stirred liquid is pumped only between the tanks, as shown in Figure 3.3.7. (a) Use the information given in the figure to construct a mathematical model for the number of pounds of salt x1(t) and x2(t) at time t in tanks A and B, respectively. (b) Find a relationship between the variables x1(t) and x2(t) that holds at time t. Explain why this relationship makes intuitive sense. Use this relationship to help find the amount of salt in tank B at t = 30 min. FIGURE 3.3.7 Mixing tanks in Problem 9 10E Consider the Lotka-Volterra predator-prey model defined by dxdt=0.1x+0.02xydydt=0.2y0.025xy, where the populations x(t) (predators) and y(t) (prey) are measured in thousands. Suppose x(0) = 6 and y(0) = 6. Use a numerical solver to graph x(t) and y(t). Use the graphs to approximate the time t 0 when the two populations are first equal. Use the graphs to approximate the period of each population.14E Determine a system of first-order differential equations that describes the currents i2(t) and i3(t) in the electrical network shown in Figure 3.3.10.16E 17E 18E 19E 20E Mixtures Solely on the basis of the physical description of the mixture problem on page 108 and in Figure 3.3.1, discuss the nature of the functions x1(t) and x2(t). What is the behavior of each function over a long period of time? Sketch possible graphs of x1(t) and x2(t). Check your conjectures by using a numerical solver to obtain numerical solution curves of (3) subject to the initial conditions x1(0) = 25, x1(0) = 0. MIXTURES Consider the two tanks shown in Figure 3.3.1. Let us suppose for the sake of discussion that tank A contains 50 gallons of water in which 25 pounds of salt is dissolved. Suppose tank B contains 50 gallons of pure water. Liquid is pumped into and out of the tanks as indicated in the figure; the mixture exchanged between the two tanks and the liquid pumped out of tank B are assumed to be well stirred. We wish to construct a mathematical model that describes the number of pounds x1(t) and x2(t) of salt in tanks A and B, respectively, at time t. Figure 3.3.1 Connected mixing tanks (3) 22E Answer Problems 1 and 2 without referring back to the text. Fill in the blank or answer true or false. 1. If P(t) = P0e0.15t gives the population in an environment at time t, then a differential equation satisfied by P(t) is __________.2RE 3RE 4RE tzi the Iceman In September of 1991 two German tourists found the well-preserved body of a man partially frozen in a glacier in the tztal Alps on the border between Austria and Italy. See Figure 3.R.1. Through the carbon-dating technique it was found that the body of tzi the icemanas he came to be calledcontained 53% as much C-14 as found in a living person. Assume that the iceman was carbon dated in 1991. Use the method illustrated in Example 3 of Section 3.1 to find the approximate date of his death.6RE 7RE 8RE 9RE According to Stefans law of radiation the absolute temperature T of a body cooling in a medium at constant absolute temperature Tm is given by dTdt=k(T4Tm4), where k is a constant. Stefans law can be used over a greater temperature range than Newtons law of cooling. (a) Solve the differential equation. (b) Show that when T Tm is small in comparison to Tm then Newton's law of cooling approximates Stefans law. [Hint: Think binomial series of the right-hand side of the DE.]11RE A classical problem in the calculus of variations is to find the shape of a curve C such that a bead, under the influence of gravity, will slide from point A(0, 0) to point B(x1, y1) in the least time. See Figure 3.R.3. It can be shown that a nonlinear differential for the shape y(x) of the path is y[1 + (y)2] = k, where k is a constant. First solve for dx in terms of y and dy, and then use the substitution y = k sin2 to obtain a parametric form of the solution. The curve C turns out to be a cycloid. FIGURE 3.R.3 Sliding bead in Problem 12 13RE 14RE 15RE 16RE 17RE 18RE

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