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All Textbook Solutions for A First Course in Differential Equations with Modeling Applications (MindTap Course List)
48E(a)With a slight change in notation the transform in (6) is the same as L{f(t)}=sL{f(t)}f(0). Withf(t)=teat, discuss how this result in conjunction with (c) of Theorem 7.1.1 can be used to evaluate L{teat}. (b)Proceed as in part (a), but this time discuss how to use (7) with f(t)=tsinkt in conjunction with (d) and (e) of Theorem 7.1.1 to evaluate L{tsinkt}.Make up two functions f1 and f2 that have the same Laplace transform. Do not think profound thoughts.51ESuppose f(t) is a function for which f (t) is piecewise continuous and of exponential order c. Use results in this section and Section 7.1 to justify f(0)=limssF(s), where F(s) = +{f(t)}. Verify this result with f(t) = cos kt.In Problems 120 find either F(s) or f(t), as indicated. 1. {te10t}In Problems 120 find either F(s) or f(t), as indicated. 2. {te6t}In Problems 120 find either F(s) or f(t), as indicated. 3. {t3e2t}In Problems 120 find either F(s) or f(t), as indicated. 4. {t10e7t}In Problems 120 find either F(s) or f(t), as indicated. 5. {t(et+e2t)2}Find either F(s) or f (t), as indicated. {e2t(t 1)2}Find either F(s) or f (t), as indicated. {et sin 3t}8EIn Problems 120 find either F(s) or f(t), as indicated. 9. {(1et+3e4t)cos5t}In Problems 120 find either F(s) or f(t), as indicated. 10. {e3t(94t+10sint2)}In Problems 120 find either F(s) or f(t), as indicated. 11. 1{1(s+2)3}In Problems 120 find either F(s) or f(t), as indicated. 12. 1{1(s1)4}In Problems 120 find either F(s) or f(t), as indicated. 13. 1{1s26s+10}In Problems 120 find either F(s) or f(t), as indicated. 14. 1{1s2+2s+5}In Problems 120 find either F(s) or f(t), as indicated. 15. 1{ss2+4s+5}In Problems 120 find either F(s) or f(t), as indicated. 16. 1{2s+5s2+6s+34}In Problems 120 find either F(s) or f(t), as indicated. 17. 1{s(s+1)2}In Problems 120 find either F(s) or f(t), as indicated. 18. 1{5s(s2)2}In Problems 120 find either F(s) or f(t), as indicated. 19. 1{2s1s2(s+1)3}In Problems 120 find either F(s) or f(t), as indicated. 20. 1{(s+1)2(s+2)4}In Problems 2130 use the Laplace transform to solve the given initial-value problem. 21. y + 4y = e4t, y(0) = 2In Problems 2130 use the Laplace transform to solve the given initial-value problem. 21. y y = 1 + tet, y(0) = 0In Problems 2130 use the Laplace transform to solve the given initial-value problem. 23. y + 2y + y = 0, y(0) = 1, y(0) = 1In Problems 2130 use the Laplace transform to solve the given initial-value problem. 24. y 4y + 4y = t3e2t, y(0) = 0, y(0) = 0In Problems 2130 use the Laplace transform to solve the given initial-value problem. 25. y 6y + 9y = t, y(0) = 0, y(0) = 1In Problems 2130 use the Laplace transform to solve the given initial-value problem. 26. y 4y + 4y = t3, y(0) = 1, y(0) = 0In Problems 2130 use the Laplace transform to solve the given initial-value problem. 27. y6y+13y=0,y(0)=0,y(0)=3In Problems 2130 use the Laplace transform to solve the given initial-value problem. 28. 2y + 20y + 51y = 0, y(0) = 2, y(0) = 0In Problems 2130 use the Laplace transform to solve the given initial-value problem. 29. y y = et cos t, y(0) = 0, y(0) = 0In Problems 2130 use the Laplace transform to solve the given initial-value problem. 30. y 2y + 5y = 1 + t, y(0) = 0, y(0) = 4In Problems 31 and 32 use the Laplace transform and the procedure outlined in Example 10 to solve the given boundary-value problem. 31. y + 2y + y = 0, y(0) = 2, y(1) = 2In Problems 31 and 32 use the Laplace transform and the procedure outlined in Example 10 to solve the given boundary-value problem. 32. y+8y+20y=0,y(0)=0,y()=0 EXAMPLE 10 A Boundary-Value Problem A beam of length L is embedded at both ends, as shown in Figure 7.3.8. Find the deflection of the beam when the load is given by w(x)={w0(12Lx),0xL/20,L/2xL.A 4-pound weight stretches a spring 2 feet. The weight is released from rest 18 inches above the equilibrium position, and the resulting motion takes place in a medium offering a damping force numerically equal to 78 times the instantaneous velocity. Use the Laplace transform to find the equation of motion x(t).Recall that the differential equation for the instantaneous charge q(t) on the capacitor in an LRC-series circuit is given by Ld2qdt2+Rdqdt+1Cq=E(t)(20) See Section 5.1. Use the Laplace transform to find q(t) when L = 1 h, R = 20 , C = 0.005 f, E(t) = 150 V, t 0, q(0) = 0, and i(0) = 0. What is the current i(t)?Consider a battery of constant voltage E0 that charges the capacitor shown in Figure 7.3.10. Divide equation (20) by L and define 2 = R/L and 2 = 1/LC. Use the Laplace transform to show that the solution q(t) of q + 2q + 2q = E0/L subject to q(0) = 0, i(0) = 0 is q(t)={E0C[1et(cosh22t+22sinh22t)],,E0C[1et(1+t)],=,E0C[1et(cos22t+22sin22t)],. Figure 7.3.10 Series circuit in Problem 35Use the Laplace transform to find the charge q(t) in an RC series circuit when q(0) = 0 and E(t) = E0ekt, k 0. Consider two cases: k 1/RC and k = 1/RC.In Problems 3748 find either F(s) or f(t), as indicated. 37. (t1)U(t1)In Problems 3748 find either F(s) or f(t), as indicated. 38. e2tU(t2)In Problems 3748 find either F(s) or f(t), as indicated. 39. tU(t2)In Problems 3748 find either F(s) or f(t), as indicated. 40. (3t+1)U(t1)In Problems 3748 find either F(s) or f(t), as indicated. 41. cos2tU(t)In Problems 3748 find either F(s) or f(t), as indicated. 42. {sintU(t2)}In Problems 3748 find either F(s) or f(t), as indicated. 43. 1{e2ss3}In Problems 3748 find either F(s) or f(t), as indicated. 44. 1{(1+e2s)2s+2}In Problems 3748 find either F(s) or f(t), as indicated. 45. 1{ess2+1}In Problems 3748 find either F(s) or f(t), as indicated. 46. 1{ses/2s2+4}In Problems 3748 find either F(s) or f(t), as indicated. 47. 1{ess(s+1)}In Problems 3748 find either F(s) or f(t), as indicated. 48. 1{e2ss2(s1)}In Problems 4954 match the given graph with one of the functions in (a)(f). The graph of f(t) is given in Figure 7.3.11. (a) f(t)f(t)u(ta) (b) f(tb)u(tb) (c) f(t)u(ta) (d) f(t)f(t)u(tb) (e) f(t)u(ta)f(t)u(tb) (f) f(ta)u(ta)f(ta)u(tb) FIGURE 7.3.11 Graph for Problems 49-54 49. FIGURE 7.3.12 Graph for Problem 49In Problems 4954 match the given graph with one of the functions in (a)(f). The graph of f(t) is given in Figure 7.3.11. (a) f(t)f(t)u(ta) (b) f(tb)u(tb) (c) f(t)u(ta) (d) f(t)f(t)u(tb) (e) f(t)u(ta)f(t)u(tb) (f) f(ta)u(ta)f(ta)u(tb) FIGURE 7.3.11 Graph for Problems 4954 50. FIGURE 7.3.13 Graph for Problem 50In Problems 4954 match the given graph with one of the functions in (a)(f). The graph of f(t) is given in Figure 7.3.11. (a) f(t)f(t)u(ta) (b) f(tb)u(tb) (c) f(t)u(ta) (d) f(t)f(t)u(tb) (e) f(t)u(ta)f(t)u(tb) (f) f(ta)u(ta)f(ta)u(tb) FIGURE 7.3.11 Graph for Problems 4954 51. Figure 7.3.14 Graph for Problem 51In Problems 4954 match the given graph with one of the functions in (a)(f). The graph of f(t) is given in Figure 7.3.11. (a) f(t)f(t)u(ta) (b) f(tb)u(tb) (c) f(t)u(ta) (d) f(t)f(t)u(tb) (e) f(t)u(ta)f(t)u(tb) (f) f(ta)u(ta)f(ta)u(tb) FIGURE 7.3.11 Graph for Problems 4954 52. Figure 7.3.15 Graph for Problem 52In Problems 4954 match the given graph with one of the functions in (a)(f). The graph of f(t) is given in Figure 7.3.11. (a) f(t)f(t)u(ta) (b) f(tb)u(tb) (c) f(t)u(ta) (d) f(t)f(t)u(tb) (e) f(t)u(ta)f(t)u(tb) (f) f(ta)u(ta)f(ta)u(tb) FIGURE 7.3.11 Graph for Problems 4954 53. Figure 7.3.16 Graph for Problem 53In Problems 4954 match the given graph with one of the functions in (a)(f). The graph of f(t) is given in Figure 7.3.11. (a) f(t)f(t)u(ta) (b) f(tb)u(tb) (c) f(t)u(ta) (d) f(t)f(t)u(tb) (e) f(t)u(ta)f(t)u(tb) (f) f(ta)u(ta)f(ta)u(tb) FIGURE 7.3.11 Graph for Problems 4954 54. FIGURE 7.3.17 Graph for Problem 54In Problems 5562 write each function in terms of unit step functions. Find the Laplace transform of the given function. 55. f(t)={2,0t32,t3In Problems 5562 write each function in terms of unit step functions. Find the Laplace transform of the given function. 56. f(t)={1,0t40,4t51,t5In Problems 5562 write each function in terms of unit step functions. Find the Laplace transform of the given function. 57. f(t)={0,0t1t2,t1In Problems 5562 write each function in terms of unit step functions. Find the Laplace transform of the given function. 58. f(t)={0,0t3/2sint,t3/2In Problems 5562 write each function in terms of unit step functions. Find the Laplace transform of the given function. 59. f(t)={t,0t20,t2In Problems 5562 write each function in terms of unit step functions. Find the Laplace transform of the given function. 60. f(t)={sint,0t20,t2In Problems 5562 write each function in terms of unit step functions. Find the Laplace transform of the given function. 61.62EIn Problems 6370 use the Laplace transform to solve the given initial-value problem. y + y = f(t), y(0) = 0, where 63. f(t)={0,0t15,t1use the Laplace transform to solve the given initial-value problem. y + y = f(t), y(0) = 0, where f(t)={1,0t11,t165Ey + 4y = f(t), y(0) = 0, y(0) = 1, where f(t)={1,0t10,t1y + 4y = sin t U(t 2), y(0) = 1, y(0) = 0use the Laplace transform to solve the given initial-value problem. y 5y + 6y = U(t 1), y(0) = 0, y(0) = 1y + y = f(t), y(0) = 0, y(0) = 1, where f(t)={0,0t1,t20,t2y+4y+3y=1U(t2)U(t4)+U(t6), y(0) = 0, y(0) = 0Suppose a 32-pound weight stretches a spring 2 feet. If the weight is released from rest at the equilibrium position, find the equation of motion x(t) if an impressed force f(t) = 20t acts on the system for 0 t 5 and is then removed (see Example 5). Ignore any damping forces. Use a graphing utility to graph x(t) on the interval [0, 10].Solve Problem 71 if the impressed force f(t) = sin t acts on the system for 0 t 2 and is then removed.In Problems 73 and 74 use the Laplace transform to find the charge q(t) on the capacitor in an RC-series circuit subject to the givenconditions. 73. q(0) = 0, R = 2.5 , C = 0.08 f, E(t) given in Figure 7.3.20 FIGURE 7.3.20 E(t) in Problem 73In Problems 73 and 74 use the Laplace transform to find the charge q(t) on the capacitor in an RC-series circuit subject to the givenconditions. 74. q(0) = q0, R = 10 , C = 0.1 f, E(t) given in Figure 7.3.21 FIGURE 7.3.21 E(t) in Problem 74(a) Use the Laplace transform to find the current i(t) in a single-loop LR-series circuit when i(0) = 0, L = 1 h, R = 10 , and E(t) is as given in Figure 7.3.22. (b) Use a graphing utility to graph i(t) for 0 t 6. Use the graph to estimate imax and imin, the maximum and minimumvalues of the current. FIGURE 7.3.22 E(t) in Problem 75(a) Use the Laplace transform to find the charge q(t) on the capacitor in an RC-series circuit when q(0) = 0, R = 50 , C = 0.01 f, and E(t) is as given in Figure 7.3.23. (b) Assume that E0 = 100 V. Use a graphing utility to graph q(t) for 0 t 6. Use the graph to estimate qmax, themaximum value of the charge. FIGURE 7.3.23 E(t) in Problem 76A cantilever beam is embedded at its left end and free at its right end. Use the Laplace transform to find the deflection y(x) when the load is given by w(x)={w0,0xL/20,L/2xL.Solve Problem 77 when the load is given by w(x)={0,0xL/3w0,L/3x2L/30,2L/3xL.Find the deflection y(x) of a cantilever beam embedded at its left end and free at its right end when the load is as given in Example 10.A beam is embedded at its left end and simply supported at its right end. Find the deflection y(x) when the load is as given in Problem 77.Discuss how you would fix up each of the following functions so that Theorem 7.3.2 could be used directly to find the given Laplacetransform. Check your answers using (16) of this section. (a) {(2t + 1) U(t 1)} (b) {et U(t 5)} (c) {cos t U(t ) (d) {(t2 3t)U(t 2)}83EIn Problems 18 use Theorem 7.4.1 to evaluate the given Laplace transform. 1. te10t2EIn Problems 18 use Theorem 7.4.1 to evaluate the given Laplace transform. 3. tcos2tIn Problems 18 use Theorem 7.4.1 to evaluate the given Laplace transform. 4. tsinh3tIn Problems 18 use Theorem 7.4.1 to evaluate the given Laplace transform. 5. t2sinhtIn Problems 18 use Theorem 7.4.1 to evaluate the given Laplace transform. 6. L{t2cost}In Problems 18 use Theorem 7.4.1 to evaluate the given Laplace transform. 7. te2tsin6tIn Problems 18 use Theorem 7.4.1 to evaluate the given Laplace transform. 8. L{te3tcos3t}In Problems 914 use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix C as needed. 9. y + y = t sin t, y(0) = 0In Problems 914 use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix C as needed. 10. yy=tetsint,y(0)=0In Problems 914 use the Laplace transform to solve the given initial- value problem. Use the table of Laplace transforms in Appendix C as needed. 11. y + 9y = cos 3t, y(0) = 2, y(0) = 5In Problems 914 use the Laplace transform to solve the given initial- value problem. Use the table of Laplace transforms in Appendix C as needed. 12. y + y = sin t, y(0) = 1, y(0) = 1In Problems 914 use the Laplace transform to solve the given initial- value problem. Use the table of Laplace transforms in Appendix C as needed. 13. y + 16y = f(t), y(0) = 0, y(0) = 1, where f(t)={cos4t,00ttIn Problems 914 use the Laplace transform to solve the given initial- value problem. Use the table of Laplace transforms in Appendix C as needed. 14. y + y = f(t), y(0) = 1, y(0) = 0, where f(t)={1,sint,0t/2t/2In Problems 15 and 16 use a graphing utility to graph the indicated solution. 15. y(t) of Problem 13 for 0 t 2In Problems 15 and 16 use a graphing utility to graph the indicated solution. 16. y(t) of Problem 14 for 0 t 3In some instances the Laplace transform can be used to solve linear differential equations with variable monomial coefficients. In Problems 17 and 18 use Theorem 7.4.1 to reduce the given differential equation to a linear first-order DE in the transformed function Y(s)=y(t). Solve the first-order DE for Y(s) and then find Y(t)=1Y(s). 17. ty y = 2t2, y(0) = 0In some instances the Laplace transform can be used to solve linear differential equations with variable monomial coefficients. In Problems 17 and 18 use Theorem 7.4.1 to reduce the given differential equation to a linear first-order DE in the transformed function Y(s)=Ly(t). Solve the first-order DE for Y(s) and then find Y(t)=L1y(s). 18. 2y + ty 2y = 10, y(0) = y(0) = 0In Problems 1922 proceed as in Example 3 and find the convolution fg of the given functions. After integrating find the Laplace transform of fg. 19. f(t)=4t,g(t)=3t2In Problems 1922 proceed as in Example 3 and find the convolution fg of the given functions. After integrating find the Laplace transform of fg. 20. f(t)=t,g(t)=etIn Problems 1922 proceed as in Example 3 and find the convolution fg of the given functions. After integrating find the Laplace transform of fg. 21. f(t)=et,g(t)=etIn Problems 1922 proceed as in Example 3 and find the convolution fg of the given functions. After integrating find the Laplace transform of fg. 22. f(t)=cos2t,g(t)=etIn Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming. 23. 1t3In Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming. 24. t2tetIn Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming. 25. etetcostIn Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming. 26. e2tsintIn Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming. 27. {0ted}In Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming. 27. {0tcosd}In Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming. 29. {0tecosd}In Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming. 30. {0tsind}In Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming. 31. {0tetd}In Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming. 32. {0tsincos(t)d}In Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming. 33. {t0tsind}In Problems 2334 proceed as in Example 4 and find the Laplace transform of f g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming. 34. {t0ted}In Problems 3538 use (8) to evaluate the given inverse transform. 35. L1{1s(s1)}In Problems 3538 use (8) to evaluate the given inverse transform. 36. L1{1s2(s1)}In Problems 3538 use (8) to evaluate the given inverse transform. 37. L1{1s3(s1)}In Problems 3538 use (8) to evaluate the given inverse transform. 38. L1{s(sa)2}Use the Laplace transform and the results of Problem 39 to solve the initial-value problem y+y=sint+tsint,y(0)=0,y(0)=0. Use a graphing utility to graph the solution.In Problems 4150 use the Laplace transform to solve the given integral equation or integrodifferential equation. 41. f(t)+0t(t)f()d=tIn Problems 4150 use the Laplace transform to solve the given integral equation or integrodifferential equation. 42. f(t)=2t40tsinf(t)dIn Problems 4150 use the Laplace transform to solve the given integral equation or integrodifferential equation. 43. f(t)=tet+0tf(t)dIn Problems 4150 use the Laplace transform to solve the given integral equation or integrodifferential equation. 44. f(t)+20tf()cos(t)d=4et+sintIn Problems 4150 use the Laplace transform to solve the given integral equation or integrodifferential equation. 45. f(t)+0tf()d=1In Problems 4150 use the Laplace transform to solve the given integral equation or integrodifferential equation. 46. f(t)=cost+0tef(t)dIn Problems 4150 use the Laplace transform to solve the given integral equation or integrodifferential equation. 47. f(t)=1+t830t(t)3f()dIn Problems 4150 use the Laplace transform to solve the given integral equation or integrodifferential equation. 48. t2f(t)=0t(ee)f(t)dIn Problems 4150 use the Laplace transform to solve the given integral equation or integrodifferential equation. 49. y(t)=1sint0ty()d, y(0) = 0In Problems 4150 use the Laplace transform to solve the given integral equation or integrodifferential equation. 50. dydt+6y(t)+90ty()d=1,y(0)=0In Problems 51 and 52 solve equation (10) subject to i(0) = 0 with L, R, C, and E(t) as given. Use a graphing utility to graph the solution for 0 t 3. 51. L=0.1h,R=3,C=0.05f,E(t)=100[U(t1)U(t2)]In Problems 51 and 52 solve equation (10) subject to i(0) = 0 with L, R, C, and E(t) as given. Use a graphing utility to graph the solution for 0 t 3. 52. L=0.005h,R=1,C=0.02f,E(t)=100[t(t1)U(t1)]In Problems 5358 use Theorem 7.4.3 to find the Laplace transform of the given periodic function. 53.In Problems 5358 use Theorem 7.4.3 to find the Laplace transform of the given periodic function. 54.In Problems 5358 use Theorem 7.4.3 to find the Laplace transform of the given periodic function. 55. FIGURE 7.4.8 Graph for Problem 55In Problems 5358 use Theorem 7.4.3 to find the Laplace transform of the given periodic function. 56.In Problems 5358 use Theorem 7.4.3 to find the Laplace transform of the given periodic function. 57.In Problems 5358 use Theorem 7.4.3 to find the Laplace transform of the given periodic function. 58.In Problems 59 and 60 solve equation (15) subject to i(0) = 0 with E(t) as given. Use a graphing utility to graph the solution for 0 t 4 in the case when L = 1 and R = 1. 59. E(t) is the meander function in Problem 53 with amplitude 1 and a = 1. The differential equation for the current i(t) in a single-loop LR-series circuit is Ldidt+Ri=E(t). (15) 53. FIGURE 7.4.6 Graph for Problem 53In Problems 59 and 60 solve equation (15) subject to i(0) = 0 with E(t) as given. Use a graphing utility to graph the solution for 0 t 4 in the case when L = 1 and R = 1. 60. E(t) is the sawtooth function in Problem 55 with amplitude 1 and b = 1.In Problems 61 and 62 solve the model for a driven spring/mass system with damping md2xdt2+dxdt+kx=f(t),x(0)=0,x(0)=0, where the driving function f is as specified. Use a graphing utility to graph x(t) for the indicated values of t. 61. m=12, = 1, k = 5, f is the meander function in Problem 53 with amplitude 10, and a = , 0 t 2.In Problems 61 and 62 solve the model for a driven spring/mass system with damping md2xdt2+dxdt+kx=f(t),x(0)=0,x(0)=0, where the driving function f is as specified. Use a graphing utility to graph x(t) for the indicated values of t. 62. m = 1, = 1, k = 1, f is the meander function in Problem 54 with amplitude 5, and a = , 0 t 4.Discuss how Theorem 7.4.1 can be used to find 1{lns3s+1}.In Section 6.4 we saw that ty + y + ty = 0 is Bessels equation of order v = 0. In view of (24) of that section and Table 6.4.1 a solution of the initial-value problem ty + y + ty = 0, y(0) = 1, y(0) = 0, is y = J0(t). Use this result and the procedure outlined in the instructions to Problems 17 and 18 to show that {J0(t)}=1s2+1. [Hint: You might need to use Problem 52 in Exercises 7.2.](a)Laguerres differential equation ty+(1t)y+ny=0 is known to possess polynomial solutions when n is a nonnegative integer. These solutions are naturally called Laguerre polynomials and are denoted by Ln(t). Find y=Ln(t), for n = 0, 1, 2, 3, 4 if it is known that Ln(0)=1. (b)Show that L{etn!dndtntnet}=Y(s), where Y(s)=L(y) and y=L(t) is a polynomial solution of the DE in part (a). Conclude that Ln(t)=etn!dndtntnet,n=0,1,2, This last relation for generating the Laguerre polynomials is the analogue of Rodrigues formula for the Legendre polynomials. See (36) in Section 6.4. Rodrigues formula for these polynomials is Pn(x)=12nn!dndxn(x21)n,n=0,1,2,(36)The Laplace transform L{et2} exists, but without finding it solve the initial-value problem y+y=et2,y(0)=0,y(0)=0.Solve the integral equation f(t)=et+et0tef()d.(a) Show that the square wave function E(t) given in Figure 7.4.4 can be written E(t)=k=0(1)ku(tk). (b) Obtain (14) of this section by taking the Laplace transform of each term in the series in part (a).Use the Laplace transform as an aide in evaluating the improper integral 0te2t sin 4t dt.70E71EAppropriately modify the procedure of Problem 72 to find a solution of y+3y4y=0,y(0)=0,y(0)=0,y(0)=1.The charge q(t) on a capacitor in an LC-series circuit is given by d2qdt2+q=14U(t)+6U(t3),q(0)=0,q(0)=0. Appropriately modify the procedure of Problem 72 to find q(t). Graph your solution.In Problems 112 use the Laplace transform to solve the given initial-value problem. 1. y 3y = (t 2), y(0) = 0In Problems 112 use the Laplace transform to solve the given initial-value problem. 2. y + y = (t 1), y(0) = 2In Problems 112 use the Laplace transform to solve the given initial-value problem. 3. y + y = (t 2), y(0) = 0, y(0) = 1In Problems 112 use the Laplace transform to solve the given initial-value problem. 4. y+16y=(t2),y(0)=0,y(0)=0In Problems 112 use the Laplace transform to solve the given initial-value problem. 5. y+y=(t12)+(t32), y(0) = 0, y(0) = 0In Problems 112 use the Laplace transform to solve the given initial-value problem. 6. y + y = (t 2) + (t 4), y(0) = 1, y(0) = 0In Problems 112 use the Laplace transform to solve the given initial-value problem. 7. y + 2y = (t 1), y(0) = 0, y(0) = 1In Problems 112 use the Laplace transform to solve the given initial-value problem. 8. y 2y = 1 + (t 2), y(0) = 0, y(0) = 1In Problems 112 use the Laplace transform to solve the given initial-value problem. 9. y + 4y + 5y = (t 2), y(0) = 0, y(0) = 010EIn Problems 112 use the Laplace transform to solve the given initial-value problem. 11. y + 4y + 13y = (t ) + (t 3), y(0) = 1, y(0) = 0In Problems 112 use the Laplace transform to solve the given initial-value problem. 12. y7y+6y=et+(t2)+(t4),y(0)=0,y(0)=0In Problems 13 and 14 use the Laplace transform to solve the given initial-value problem. Graph your solution on the interval [0, 8]. 13. y+y=k=1(tk),y(0)=0,y(0)=1In Problems 13 and 14 use the Laplace transform to solve the given initial-value problem. Graph your solution on the interval [0, 8]. 14. y+y=k=1(t2k),y(0)=0,y(0)=1In Problems 15 and 16 a uniform beam of length L carries a concentrated load w0 at x=12L. See Figure 7.5.5 (Problem 15) and Figure 7.5.6 (Problem 16). Use the Laplace transform to solve the differential equation EId4ydx4=w0(x12L),0xL, subject to the given boundary conditions. 15. y(0)=0,y(0)=0,y(L)=0,y(L)=0 FIGURE 7.5.5 Beam embedded at its left end and free at its right endIn Problems 15 and 16 a uniform beam of length L carries a concentrated load w0 at x=12L. See Figure 7.5.5 (Problem 15) and Figure 7.5.6 (Problem 16). Use the Laplace transform to solve the differential equation EId4ydx4=w0(x12L),0xL, subject to the given boundary conditions. 16. y(0)=0,y(0)=0,y(L)=0,y(L)=0 FIGURE 7.5.6 Beam embedded at both endsSomeone tells you that the solutions of the two IVPs y+2y+10y=0,y(0)=0,y(0)=1y+2y+10y=(t),y(0)=0,y(0)=0 are exactly the same. Do you agree or disagree? Defend your answer.Reread (i) in the Remarks at the end of this section. Then use the Laplace transform to solve the initial-value problem: y + 4y + 3y = et(t 1), y(0) = 0, y(0) = 2. Use a graphing utility to graph y(t) for 0 t 5.In Problems 112 use the Laplace transform to solve the given system of differential equations. 1. dxdt=x+ydydt=2xx(0)=0,y(0)=1In Problems 112 use the Laplace transform to solve the given system of differential equations. 2. dxdt=2y+etdydt=8xtx(0)=1,y(0)=13E4EIn Problems 112 use the Laplace transform to solve the given system of differential equations. 2dxdt+dydt2x =1dxdt+dydt3x3y=2 x(0) = 0, y(0) = 06Ed2xdt2+xy=0d2ydt2+yx=0 x(0) = 0, x(0) = 2, y(0) = 0, y(0) = 18E9E10E11EIn Problems 112 use the Laplace transform to solve the given system of differential equations. 12. dxdt=4x2y+2U(t1)dydt=3xy+U(t1)x(0)=0,y(0)=1213EDerive the system of differential equations describing the straight-line vertical motion of the coupled springs shown in Figure 7.6.6. Use the Laplace transform to solve the system when k1 = 1, k2 = 1, k3 = 1, m1 = 1, m2 = 1 and x1(0) = 0, x1(0)=1, x2(0) = 0, x2(0)=1. FIGURE 7.6.6 Coupled springs in Problem 14(a) Show that the system of differential equations for the currents i2(t) and i3(t) in the electrical network shown Figure 7.6.7 is L1di2dt+Ri2+Ri3=E(t)L2di3dt+Ri2+Ri3=E(t). (b) Solve the system in part (a) if R = 5 , L1 = 0.01 h, L2 = 0.0125 h, E = 100 V, i2(0) = 0, and i3(0) = 0. (c) Determine the current i1(t). FIGURE 7.6.7 Network in Problem 15(a) In Problem 12 in Exercises 3.3 you were asked to show that the currents i2(t) and i3(t) in the electrical network shown in Figure 7.6.8 satisfy Ldi2dt+Ldi3dt+R1i2=E(t) Rdi2dt+R2di3dt+1Ci3=0. Solve the system if R1 = 10 , R2 = 5 , L = 1 h, C = 0.2 f, E(t)={120,0t20,t2, i2(0) = 0, and i3(0) = 0. (b) Determine the current i1(t). FIGURE 7.6.8 Network in Problem 1617E18E19E(a) Show that the system of differential equations for the charge on the capacitor q(t) and the current i3(t) in the electrical network shown in Figure 7.6.9 is R1dqdt+1Cq+R1i3=E(t)Ldi3dt+R2i31Cq=0. (b) Find the charge on the capacitor when L = 1 h, R1 = 1 , R2 = 1 , C = 1 f, E(t)={0,0t150et,t1, i3(0) = 0, and q(0) = 0.In Problems 1 and 2 use the definition of the Laplace transform to find f(t). 1. f(t)={t,0t12t,t1In Problems 1 and 2 use the definition of the Laplace transform to find f(t). 2. f(t)={0,0t21,2t40,t43RE4REIn Problems 324 fill in the blanks or answer true or false. 5. F(s)=s2/(s2+1) is not the Laplace transform of a function that is piecewise continuous and of exponential order. _____6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24REIn Problems 25–28 use the unit step function to find an equation for each graph in terms of the function y = f(t), whose graph is given in Figure 7.R.1.
25.
FIGURE 7.R.2 Graph for Problem 25
In Problems 2528 use the unit step function to find an equation for each graph in terms of the function y = f(t), whose graph is given in Figure 7.R.1. 26. Figure 7.R.3 Graph for Problem 2627RE28REIn Problems 29–32 express f in terms of unit step functions. Find and .
29. FIGURE 7.R.6 Graph for Problem 29
30RE31REIn Problems 2932 express f in terms of unit step functions. Find f(t) and etf(t). 32. Figure 7.R.9 Graph for Problem 3233RE34RE35REIn Problems 3542 use the Laplace transform to solve the given equation. 36. y 8y + 20y = tet, y(0) = 0, y(0) = 037REIn Problems 3542 use the Laplace transform to solve the given equation. 38. y 5y = f(t), where f(t)={t2,0,0t1t1y(0)=1y + 2y = f(t), y(0) = 1, where f(t) is given in Figure 7.R.10. FIGURE 7.R.10 Graph for Problem 3940RE41REIn Problems 3542 use the Laplace transform to solve the given equation. 42. 0tf()f(t)d=6t343RE44REThe current i(t) in an RC-series circuit can be determined from the integral equation Ri+1C0ti()d=E(t), where E(t) is the impressed voltage. Determine i(t) when R = 10, C = 0.5 f, and E(t)=2(t2+t).A series circuit contains an inductor, a resistor, and a capacitor for which L=12h, R = 10 , and C = 0.01 f, respectively. The voltage E(t)={10,0,0t5t5 is applied to the circuit. Determine the instantaneous charge q(t) on the capacitor for t 0 if q(0) = 0 and q(0) = 0.A uniform cantilever beam of length L is embedded at its left end (x = 0) and free at its right end. Find the deflection y(x) if the load per unit length is given by L(x)=2w0L[L2x+(xL2)u(xL2)].When a uniform beam is supported by an elastic foundation, the differential equation for its deflection y(x) is EId4ydx4+ky=w(x), where k is the modulus of the foundation and ky is the restoring force of the foundation that acts in the direction opposite to that of the load w(x). See Figure 7.R.11. For algebraic convenience suppose that the differential equation is written as d4ydx4+4a4y=w(x)EI, where a=(k/4EI)1/4. Assume L= and a = 1. Find the deflection y(x) of a beam that is supported on an elastic foundation when (a) the beam is simply supported at both ends and a constant load w0 is uniformly distributed along its length. (b) the beam is embedded at both ends and w(x) is a concentrated load w0 applied at x=/2. [Hint: In both parts of this problem use the table of Laplace transforms in Appendix C and the fact that s4+4=(s22s+2)(s2+2s+2).]Suppose two identical pendulums are coupled by means of a spring with constant k. See Figure 7.R.12. Under the same assumptions made in the discussion preceding Example 3 in Section 7.6, it can be shown that when the displacement angles θ1(t) and θ2(t) are small, the system of linear differential equations describing the motion is
Use the Laplace transform to solve the system when θ1(0) = θ0, (0) = 0, θ2(0) = Ψ0, (0) = 0, where θ0 and Ψ0 are constants. For convenience let ω2 = g/l, K = k/m.
Use the solution in part (a) to discuss the motion of the coupled pendulums in the special case when the initial conditions are θ1(0) = θ0, (0) = 0, θ2(0) = θ0, (0) = 0. When the initial conditions are θ1(0) = θ0, (0) = 0, θ2(0) = − θ0, (0) = 0.
Coulomb Friction Revlslted In Problem 27 in Chapter 5 in Review we examined a spring/mass system in which a mass m slides over a dry horizontal surface whose coefficient of kinetic friction is a constant . The constant retarding force fk = mg of the dry surface that acts opposite to the direction of motion is called Coulomb friction after the French physicist Charles-Augustin de Coulomb (17361806). You were asked to show that the piecewise-linear differential equation for the displacement x(t) of the mass is given by md2xdt2+kx={fk,x0(motiontoleft)fk,x0(motiontoright). (a) Suppose that the mass is released from rest from a point x(0) = x0 0 and that there are no other external forces. Then the differential equations describing the motion of the mass m are x + 2x = F, 0 t T/2 x + 2x = F, T/2 t T x + 2x = F, T t 3T/2, and so on, where 2 = k/m, F = fk/m = g, g = 32, and T = 2/. Show that the times 0, T/2, T, 3T/2, ... correspond to x(t) = 0. (b) Explain why, in general, the initial displacement must satisfy 2 |x0| F. (c) Explain why the interval F/2 x F/2 is appropriately called the dead zone of the system. (d) Use the Laplace transform and the concept of the meander function to solve for the displacement x(t) for t 0. (e) Show that in the case m = 1, k = 1, fk = 1, and x0 = 5.5 that on the interval [0, 2) your solution agrees with parts (a) and (b) of Problem 28 in Chapter 5 in Review. (f) Show that each successive oscillation is 2F/2 shorter than the preceding one. (g) Predict the long-term behavior of the system.Range of a ProjectileNo Air Resistance (a) A projectile, such as the canon ball shown in Figure 7.R.13, has weight w = mg and initial velocity v0 that is tangent to its path of motion. If air resistance and all other forces except its weight are ignored, we saw in Problem 23 of Exercises 4.9 that motion of the projectile is described by the system of linear differential equations md2xdt2=0md2ydt2=mg. Use the Laplace transform to solve this system subject to the initial conditions x(0)=x,x(0)=v0cos,y(0)=0,y(0)=v0sin, where v0 = |v0| is constant and is the constant angle of elevation shown in Figure 7.R.13 on page 330. The solutions x(t) and y(t) are parametric equations of the trajectory of the projectile. (b) Use x(t) in part (a) to eliminate the parameter t in y(t). Use the resulting equation for y to show that the horizontal range R of the projectile is given by R=v02gsin2. (c) From the formula in part (b), we see that R is a maximum when sin 2 = 1 or when = /4. Show that the same rangeless than the maximumcan be attained by firing the gun at either of two complementary angles and /2 . The only difference is that the smaller angle results in a low trajectory whereas the larger angle gives a high trajectory. (d) Suppose g = 32 ft/s2, = 38, and v0 = 300 ft/s. Use part (b) to find the horizontal range of the projectile. Find the time when the projectile hits the ground. (e) Use the parametric equations x(t) and y(t) in part (a) along with the numerical data in part (d) to plot the ballistic curve of the projectile. Repeat with = 52 and v0 = 300 ft/s. Superimpose both curves on the same coordinate system. FIGURE 7.R.13 Projectile in Problem 51In Problems 16 write the given linear system in matrix form. 1. dxdt=3x5ydydt=4x+8yIn Problems 16 write the given linear system in matrix form. 2. dxdt=4x7ydydt=5xIn Problems 16 write the given linear system in matrix form. 3. dxdt=3x+4y9zdydt=6xydzdt=10x+4y+3zIn Problems 16 write the given linear system in matrix form. 4. dxdt=xydydt=x+2zdzdt=x+zIn Problems 16 write the given linear system in matrix form. 5. dxdt=xy+z+t1dydt=2x+yz3t2dzdt=x+y+z+t2t+26EIn Problems 710 write the given linear system without the use of matrices. 7. X=(4213)X+(11)etIn Problems 710 write the given linear system without the use of matrices. 8. X=(759411023)X+(021)e5t(803)e2tIn Problems 16 write the given linear system in matrix form. 9. ddt(xyz)=(112341256)(xyz)+(122)et(311)t10EIn Problems 1116 verify that the vector X is a solution of the given homogeneous linear system. 11. dxdt=3x4ydydt=4x7y;X=(12)e5t12EIn Problems 1116 verify that the vector X is a solution of the given homogeneous linear system. 13. X=(11411)X;X=(12)e3t/2In Problems 1116 verify that the vector X is a solution of the given homogeneous linear system. 14. X=(2110)X;X=(13)et+(44)tetIn Problems 1116 verify that the vector X is a solution of the given homogeneous linear system. 15. X=(121610121)X;X=(1613)In Problems 1116 verify that the vector X is a solution of the given homogeneous linear system. 16. X=(101110201)X;X=(sint12sint12costsint+cost)In Problems 1720 the given vectors are solutions of a system X = AX. Determine whether the vectors form a fundamental set on the interval (, ). 17. X1=(11)e2t,X2=(11)e6tIn Problems 1720 the given vectors are solutions of a system X = AX. Determine whether the vectors form a fundamental set on the interval (, ). 18. X1=(11)et,X2=(26)et+(88)tetIn Problems 1720 the given vectors are solutions of a system X = AX. Determine whether the vectors form a fundamental set on the interval (, ). 19. X1=(124)+t(122),X2=(124)X3=(3612)+t(244)In Problems 1720 the given vectors are solutions of a system X = AX. Determine whether the vectors form a fundamental set on the interval (, ). 20. X1=(1613),X2=(121)e4t,X3=(232)e3tIn Problems 2124 verify that the vector Xp is a particular solution of the given nonhomogeneous linear system. 21. dxdt=x+4y+2t7dydt=3x+2y4t18;Xp=(21)t+(51)In Problems 2124 verify that the vector Xp is a particular solution of the given nonhomogeneous linear system. 22. X=(2111)X+(52);Xp=(13)In Problems 2124 verify that the vector Xp is a particular solution of the given nonhomogeneous linear system. 23. X=(2134)X(17)et;Xp=(11)et+(11)tetIn Problems 2124 verify that the vector Xp is a particular solution of the given nonhomogeneous linear system. 24. X=(123420610)X+(143)sin3t;Xp=(sin3t0cos3t)Prove that the general solution of the homogeneous linear system x=(060101110)x on the interval (, ) is x=c1(615)et+c2(311)e2t+c3(211)e3t.Prove that the general solution of the nonhomogeneous linear system x=(1111)x+(11)t2+(46)t+(15) on the interval (, ) is x=c1(112)e2t+c2(11+2)e2t+(10)t2+(24)t+(10).In Problems 112 find the general solution of the given system. 1. dxdt=x+2ydydt=4x+3yIn Problems 112 find the general solution of the given system. 2. dxdt=2x+2ydydt=x+3yIn Problems 112 find the general solution of the given system. 3. dxdt=4x+2ydydt=52x+2y4EIn Problems 112 find the general solution of the given system. 5. X=(105812)XIn Problems 112 find the general solution of the given system. 6. X=(6231)XDistinct Real Eigenvalues In Problems 112 find the general solution of the given system. 7. dxdt=x+yzdydt=2ydzdt=yzDistinct Real Eigenvalues In Problems 112 find the general solution of the given system. 8. dxdt=2x7ydydt=5x+10y+4zdzdt=5y+2z9E10EDistinct Real Eigenvalues In Problems 112 find the general solution of the given system. 11. X=(11034323181412)XDistinct Real Eigenvalues In Problems 1-12 find the general solution of the given system. 12. X=(142412006)XIn Problems 13 and 14 solve the given initial-value problem. 13. X=(120112)X,X(0)=(35)In Problems 13 and 14 solve the given initial-value problem. 14. X(114020111)X,X(0)=(130)In Problem 27 of Exercises 4.9 you were asked to solve the following linear system dx1dt=150x1dx2dt=150x1275x2dx3dt=275x2125x3 using elimination techniques. This system is a mathematical model for the number of pounds of salt x1(t), x2(t), and x3(t) in the connected mixing tanks A, B, and C shown in Figure 3.3.8 on page 112. (a) Use the eigenvavalue method of this section to solve the system subject to x1(0) = 15, x2(0) = 10, x3(0) = 5. (b) What are limtx1(t), limtx2(t), and limtx3(t)? Interpret this result.(a) Use computer software to obtain the phase portrait of the system in Problem 5. If possible, include arrowheads as in Figure 8.2.2. Also include four half-lines in your phase rportrait. (b) Obtain the Cartesian equations of each of the four half-lines in part (a). (c) Draw the eigenvectors on your phase portrait of the system. 5. X = (105812) XFind phase portraits for the systems in Problems 2 and 4. For each system find any half-line trajectories and include these lines in your phase portrait. 2. dxdt=2x+2ydydt=x+3y 4. dxdt=52x+2ydydt=34x2yIn Problems 2130 find the general solution of the given system. 21. dxdt=3xydydt=9x3yIn Problems 2130 find the general solution of the given system. 22. dxdt=6x+5ydydt=5x+4yIn Problems 2130 find the general solution of the given system. 23. X=(1335)X24E25E26EIn Problems 2130 find the general solution of the given system. 27. X=(540102025)XIn Problems 2130 find the general solution of the given system. 28. X=(100031011)XIn Problems 2130 find the general solution of the given system. 29. X=(100221010)X