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All Textbook Solutions for Differential Equations with Boundary-Value Problems (MindTap Course List)
16REA mass weighing 4 pounds stretches a spring 18 inches. A periodic force equal to f(t) = cos t + sin t is impressed on the system starting at t = 0. In the absence of a damping force, for what value of will the system be in a state of pure resonance?Find a particular solution for x + 2x + 2x = A, where A is a constant force.19RE(a) A mass weighing W pounds stretches a spring 12 foot and stretches a different spring 14 foot. The two springs are attached in series and the mass is then attached to the double spring as shown in Figure 5.1.6. Assume that the motion is free and that there is no damping force present. Determine the equation of motion if the mass is initially released at a point 1 foot below- the equilibrium position with a downward velocity of 23 ft/s. (b) Show that the maximum speed of the mass is 233g+1.A series circuit contains an inductance of L= 1 h, a capacitance of C = 104 f, and an electromotive force of E(t) = 100 sin 50t V. Initially, the charge q and current i are zero. (a) Determine the charge q(t). (b) Determine the current i(t). (c) Find the times for which the charge on the capacitor is zero.22REConsider the boundary-value problem y+y=0,y(0)=y(2),y(0)=y(2). Show that except for the case = 0, there are two independent eigenfunctions corresponding to each eigenvalue.25RE26RESuppose the mass m in the spring/mass system in Problem 25 slides over a dry surface whose coefficient of sliding friction is 0. If the retarding force of kinetic friction has the constant magnitude fk = mg, where mg is the weight of the mass, and acts opposite to the direction of motion, then it is known as coulomb friction. By using the signum function sgn(x)={1,1,x0(motiontoleft)x0(motiontoleft) determine a piecewise-defined differential equation for the displacement x(t) of the damped sliding mass. 25. Suppose a mass m lying on a flat dry frictionless surface is attached to the free end of a spring whose constant is k. In Figure 5.R.2(a) the mass is shown at the equilibrium position x 5 0, that is, the spring is neither stretched nor compressed. As shown in Figure 5.R.2(b), the displacement x(t) of the mass to the right of the equilibrium position is positive and negative to the left. Determine a differential equation for the displacement x(t) of the freely sliding mass. Discuss the difference between the derivation of this DE and the analysis leading to (1) of Section 5.1. Figure 5.R.2 Sliding spring/mass system in Problem 2528RE29RESpring pendulum The rotational form of Newtons second law of motion is: The time rate of change of angular momentum about a point is equal to the moment of the resultant force (torque). In the absence of damping or other external forces, an analogue of (14) in Section 5.3 for the pendulum shown in Figure 5.3.3 is then ddt(ml2ddt)=mglsin.(1) (a) When m and l are constant show that (1) reduces to (6) of Section 5.3. (b) Now suppose the rod in Figure 5.3.3 is replaced with a spring of negligible mass. When a mass m is attached to its free end the spring hangs in the vertical equilibrium position shown in Figure 5.R.4 and has length l0. When Figure 5.R.4 Spring pendulum in Problem 30 the spring pendulum is set in motion we assume that the motion takes place in a vertical plane and the spring is stiff enough not to bend. For t 0 the length of the spring is then l(t) = l0 + x(t), where x is the displacement from the equilibrium position. Find the differential equation for the displacement angle (t) defined by (1).31REGalloping Gertie Bridges are good examples of vibrating mechanical systems that are constantly subjected to externalforces, from cars driving on them, water pushing against theirfoundation, and wind blowing across their superstructure. OnNovember 7, 1940, only four months after its grand opening,the Tacoma Narrows Suspension Bridge at Puget Sound inthe state of Washington collapsed during a windstorm. SeeFigure 5.R.6. The crash came as no surprise since GallopingGertie, as the bridge was called by local residents, was famousfor a vertical undulating motion of its roadway which gavemany motorists a very exciting crossing. For many years itwas conjectured that the poorly designed superstructure of thebridge caused the wind blowing across it to swirl in a periodicmanner and that when the frequency of this force approachedthe natural frequency of the bridge, large upheavals of thelightweight roadway resulted. In other words, it was thought thebridge was a victim of mechanical resonance. But as we haveseen on page 207, resonance is a linear phenomenon whichcan occur only in the complete absence of damping. In recentyears the resonance theory has been replaced with mathematicalmodels that can describe large oscillations even in the presenceof damping. In his project, The Collapse of the TacomaNarrows Suspension Bridge, that appeared in the last edition of this text, Gilbert N. Lewis examines simple piecewise-defined models describing the driven oscillations of a mass (a portionof the roadway) attached to a spring (a vertical support cable)for which the amplitudes of oscillation increase over time. Inthis problem you are guided through the solution of one of themodels discussed in that project. The differential equation with a piecewise-defined restoring force, d2xdt2+F(x)=sin4t,F(x)={4x,x0x, x0 is a model for the displacement x(t) of a unit mass in a driven spring/mass system. As in Section 5.1 we assume that themotion takes place along a vertical line, the equilibriumposition is x = 0, and the positive direction is downward. The restoring force acts opposite to the direction of motion: a restoring force 4x when the mass is below (x 0) theequilibrium position and a restoring force x when the mass isabove (x 0) the equilibrium position. (a) Solve the initial-value problem d2xdt2+4x=sin4t,x(0)=0,x(0)=v00. (2) The initial conditions indicate that the mass is released from the equilibrium position with a downward velocity.Use the solution to determine the first time t1 0 whenx(t) = 0, that is, the first time that the mass returns to theequilibrium position after release. The solution of (2) is defined on the interval [0, t1]. [Hint: The double-angle formulasin 4t = 2 sin 2t cos 2t will be helpful.] (b) For a time interval on which t t1 the mass is above theequilibrium position and so we must now solve the newdifferential equation d2xdt2+x=sin4t (3) One initial condition is x(t1) = 0. Find x'(t1) using the solution of (2) in part (a). Find a solution of equation (3) subject to these new initial conditions. Use the solution to determine the second time t2 t1 when x(t) = 0. The solution of (3) isdefined on the interval [t1, t2]. [Hint: Use the double-angleformula for the sine function twice.] (c) Construct and solve another initial-value problem to findx(t) defined on the interval [t2, t3], where t3 t2 is the thirdtime when x(t) = 0. (d) Construct and solve another initial-value problem to findx(t) defined on the interval [t3, t4], where t4 t3 is thefourth time when x(t) = 0. (e) Because of the assumption that v0 0 one down-upcycle of the mass is completed on the intervals [0, t2], [t2, t4], [t4, t6], and so on. Explain why the amplitudes of oscillation of the mass must increase over time. [Hint: Examine the velocity of the mass at the beginningof each cycle.] (f) Assume in (2) that v0 = 0.01. Use the four solutions on theintervals in parts (a), (b), (c), and (d) to construct a continuouspiecewise-defined function x(t) defined on the interval [0, t4].Use a graphing utility to obtain a graph of x(t) on [0, t4]. FIGURE 5.R.6 Collapse of the Tacoma Narrows Suspension BridgeIn Problems 110 find the interval and radius of convergence for the given power series. 1. n=1(1)nnxnIn Problems 110 find the interval and radius of convergence for the given power series. 2. n=11n2xn3E4E5E6E7EIn Problems 110 find the interval and radius of convergence for the given power series. 8. k=03k(4x5)kIn Problems 110 find the interval and radius of convergence for the given power series. 9. k=125k52k(x3)k10E11EIn Problems 1116 use an appropriate series in (2) to find the Maclaurin series of the given function. Write your answer in summation notation. 12. xe3x13EIn Problems 1116 use an appropriate series in (2) to find the Maclaurin series of the given function. Write your answer in summation notation. 14. x1+x2In Problems 1116 use an appropriate series in (2) to find the Maclaurin series of the given function. Write your answer in summation notation. 15. ln(1 x)16EIn Problems 17 and 18 use an appropriate series in (2) to find the Taylor series of the given function centered at the indicated value of a. Write your answer in summation notation. 17. sin x, a = 2 [Hint: Use periodicity.]18E19E20E21E22E23E24E25EIn Problems 2530 proceed as in Example 3 to rewrite the given expression using a single power series whose general term involves xk. 26. n=1ncnxn1+3n=0cnxn+2In Problems 2530 proceed as in Example 3 to rewrite the given expression using a single power series whose general term involves xk. 27. n=12ncnxn1+n=06cnxn+128EIn Problems 2530 proceed as in Example 3 to rewrite the given expression using a single power series whose general term involves xk. 29. n=2n(n1)cnxn22n=1ncnxn+n=0cnxnIn Problems 2530 proceed as in Example 3 to rewrite the given expression using a single power series whose general term involves xk. 30. n=2n(n1)cnxn+2n=2n(n1)cnxn2+3n=1ncnxnIn Problems 3134 verify by direct substitution that the given power series is a solution of the indicated differential equation. [Hint: For a power x2n+1 let k = n + 1.] 31. y=n=0(1)nn!x2n, y + 2xy = 0In Problems 3134 verify by direct substitution that the given power series is a solution of the indicated differential equation. [Hint: For a power x2n+1 let k = n + 1.] 32. y=n=0(1)nx2n, (1 + x2)y + 2xy = 0In Problems 3134 verify by direct substitution that the given power series is a solution of the indicated differential equation. [Hint: For a power x2n+1 let k = n + 1.] 33. y=n=1(1)n+1nxn, (x + 1)y + y = 034E35E36E37EIn Problems 3538 proceed as in Example 4 and find a power series solution y=n=0cnxn of the grim linear first order differential equation. 38. (1 + x)y + y = 039E40EIn Problems 1 and 2 without actually solving the given differential equation, find the minimum radius of convergence of power series solutions about the ordinary point x = 0. About the ordinary point x = 1. 1. (x2 25)y + 2xy + y = 02EIn Problems 3–6 find two power series solutions of the given differential equation about the ordinary point x = 0. Compare the series solutions with the solutions of the differential equations obtained using the method of Section 4.3. Try to explain any differences between the two forms of the solutions.
3. y″ + y = 0
In Problems 36 find two power series solutions of the given differential equation about the ordinary point x = 0. Compare the series solutions with the solutions of the differential equations obtained using the method of Section 4.3. Try to explain any differences between the two forms of the solutions. 4. y y = 0In Problems 36 find two power series solutions of the given differential equation about the ordinary point x = 0. Compare the series solutions with the solutions of the differential equations obtained using the method of Section 4.3. Try to explain any differences between the two forms of the solutions. 5. y y = 0In Problems 36 find two power series solutions of the given differential equation about the ordinary point x = 0. Compare the series solutions with the solutions of the differential equations obtained using the method of Section 4.3. Try to explain any differences between the two forms of the solutions. 6. y + 2y = 07E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23EIn Problems 23 and 24 use the procedure in Example 8 to find two power series solutions of the given differential equation about the ordinary point x = 0. 24. y + exy y = 0Without actually solving the differential equation (cos x)y + y + 5y = 0, find the minimum radius of convergence of power series solutions about the ordinary point x = 0. About the ordinary point x = 1.How can the power series method be used to solve the nonhomogeneous equation yn xy = 1 about the ordinary point x = 0? Of y 4xy 4y = ex? Carry out your ideas by solving both DEs.27E28EIn Problems 110 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. 1. x3y + 4x2y + 3y = 0In Problems 110 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. 2. x(x + 3)2y y = 03E4E5E6E7EIn Problems 110 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. 8. x(x2 + 1)2y + y = 09EIn Problems 110 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. 10. (x2 2x2 + 3x)y + x(x 3)y (x + 1)y = 011E12EIn Problems 13 and 14, x = 0 is a regular singular point of the given differential equation. Use the general form of the indicial equation in (14) to find the indicial roots of the singularity. Without solving, discuss the number of series solutions you would expect to find using the method of Frobenius. 13. x2y+(53x+x2)y13y=014EIn Problems 1524, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, ). 15. 2xy y + 2y = 016E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E35E36E37E1E2E3EBessels Equation In Problems 16 use (1) to find the general solution of the given differential equation on (0, ). 4. 16x2y + 16xy + (16x2 1)y = 0Bessels Equation In Problems 16 use (1) to find the general solution of the given differential equation on (0, ). 5. xy + y + xy = 0Bessels Equation In Problems 16 use (1) to find the general solution of the given differential equation on (0, ). 6. ddx[xy]+(x4x)y=07E8E9E10EIn Problems 11 and 12 use the indicated change of variable to find the general solution of the given differential equation on (0, ). 11. x2y + 2xy + 2x2y = 0; y = x1/2v(x)In Problems 11 and 12 use the indicated change of variable to find the general solution of the given differential equation on (0, ). 12. x2y+(2x2v2+14)y=0;y=xv(x)13E14E15E16E17E18E19EIn Problems 1320 use (20) to find the general solution of the given differential equation on (0, ). 20. 9x2y+9xy+(x636)y=021EAssume that b in equation (20) can be pure imaginary, that is, b = i, 0, i2 = 1. Use this assumption to express the general solution of the given differential equation in terms of the modified Bessel functions In and Kn. (a) y x2y = 0 (b) xy + y 7x3y = 023E24EIn Problems 2326 first use (20) to express the general solution of the given differential equation in terms of Bessel functions. Then use (26) and (27) to express the general solution in terms of elementary functions. 25. 16x2y + 32xy + (x4 12)y = 0In Problems 2326 first use (20) to express the general solution of the given differential equation in terms of Bessel functions. Then use (26) and (27) to express the general solution in terms of elementary functions. 26. 4x2y 4xy + (16x2 + 3)y = 027E28E29E30E31EUse the recurrence relation in Problem 28 along with (26) and (27) to express J3/2(x), J3/2(x), J5/2(x), and J5/2(x) in terms of sin x, cos x, and powers of x.33E34EUse the change of variables s=2kmet/2 to show that the differential equation of the aging spring mx + ketx = 0, 0, becomes s2d2xds2+sdxds+s2x=0.36EUse the result in parts (a) and (b) of Problem 36 to express the general solution on (0, ) of each of the two forms of Airys equation in terms of Bessel functions. 36. Show that y=x1/2w(23x3/2) is a solution of the given form of Airys differential equation whenever w is a solution of the indicated Bessels equation. [Hint: After differentiating, substituting, and simplifying, then let t=23x3/2.] (a) y+2xy=0,x0;t2w+tw+(t219)w=0,t0 (b) y2xy=0,x0;t2w+tw(t2+19)w=0,t038E39E(a) Use the explicit solutions y1(x) and y2(x) of Legendres equation given in (32) and the appropriate choice of c0 and c1 to find the Legendre polynomials P6(x) and P7(x). (b) Write the differential equations for which P6(x) and P7(x) are particular solutions.47E48EFind the first three positive values of for which the problem (1 x2)y 2xy + y = 0, y(0) = 0, y(x), y(x) bounded on [1, 1]The differential equation y 2xy + 2y = 0 is known as Hermites equation of order after the French mathematician Charles Hermite (18221901). Show that the general solution of the equation is y(x) = c0y1(x) + c1y2(x), where y1(x)=1+k=1(1)k2k(2)(2k+2)(2k)!x2ky2(x)=x+k=1(1)k2k(1)(3)(2k+1)(2k+1)!x2k+1 are power series solutions centered at the ordinary point 0.(a) When = n is a nonnegative integer, Hermites differential equation always possesses a polynomial solution of degree n. Use y1(x), given in Problem 53, to find polynomial solutions for n = 0, n = 2, and n = 4. Then use y2(x) to find polynomial solutions for n = 1, n = 3, and n = 5. (b) A Hermite polynomial Hn(x) is defined to be the nth degree polynomial solution of Hermites equation multiplied by an appropriate constant so that the coefficient of xn in Hn(x) is 2n. Use the polynomial solutions in part (a) to show that the first six Hermite polynomials are H0(x)=1H1(x)=2xH2(x)=4x22H3(x)=8x312xH4(x)=16x448x2+12H5(x)=32x5160x3+120x.55E56EIn Problems 1 and 2 answer true or false without referring back to the text. 1. The general solution of x2y + xy + (x2 1)y = 0 is y = c1J1(x) + c1J1 (x) ________2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25REExpress the general solution of the given differential equation on the interval (0, ) in terms of Bessel functions. (a) 4x2y + 4xy + (64x2 25)y = 0 (b) x2y + xy (18x2 + 9)y = 027RE28REIn Problems 118 use Definition 7.1.1 to find f(t). 1. f(t)={1,0t11,t1In Problems 118 use Definition 7.1.1 to find f(t). 2. f(t)={4,0t20,t2In Problems 118 use Definition 7.1.1 to find f(t). 3. f(t)={t,0t11,t14E5E6E7EIn Problems 118 use Definition 7.1.1 to find f(t). 8.In Problems 118 use Definition 7.1.1 to find f(t). 9.10E11E12EIn Problems 118 use Definition 7.1.1 to find f(t). 13. f(t) = te4t14E15E16EIn Problems 118 use Definition 7.1.1 to find f(t). 17. f(t) = t cos tIn Problems 118 use Definition 7.1.1 to find f(t). 18. f(t) = t sin tIn Problems 1936 use Theorem 7.1.1 to find f(t). 19. f(t) = 2t420E21EIn Problems 1936 use Theorem 7.1.1 to find f(t). 22. f(t) = 7t + 3In Problems 1936 use Theorem 7.1.1 to find f(t). 23. f(t) = t2 + 6t 324EIn Problems 1936 use Theorem 7.1.1 to find f(t). 25. f(t) = (t + 1)3In Problems 1936 use Theorem 7.1.1 to find f(t). 26. f(t) = (2t 1)327E28EIn Problems 1936 use Theorem 7.1.1 to find f(t). 29. f(t) = (1 + e2t)2In Problems 1936 use Theorem 7.1.1 to find f(t). 30. f(t) = (et et)2In Problems 1936 use Theorem 7.1.1 to find f(t). 31. f(t) = 4t2 5 sin 3tIn Problems 1936 use Theorem 7.1.1 to find f(t). 32. f(t) = cos 5t + sin 2tIn Problems 1936 use Theorem 7.1.1 to find f(t). 33. f(t) = sinh ktIn Problems 1936 use Theorem 7.1.1 to find f(t). 34. f(t) = cosh ktIn Problems 1936 use Theorem 7.1.1 to find f(t). 35. f(t) = et sinh tIn Problems 1936 use Theorem 7.1.1 to find f(t). 36. f(t) = et cosh tIn Problems 3740 find f(t) by first using a trigonometric identity. 37. f(t) = sin 2t cos 2tIn Problems 3740 find f(t) by first using a trigonometric identity. 38. f(t) = cos2tIn Problems 3740 find f(t) by first using a trigonometric identity. 39. f(t) = sin(4t + 5)In Problems 3740 find f(t) by first using a trigonometric identity. 40. f(t)=10cos(t6)41E42E43E44EIn Problems 4346 use Problems 41 and 42 and the fact that (12)= to find the Laplace transform of the given function. 45. f(t) = t3/246ESuppose that if {f1(t)} = F1(s) for s c1 and that {f2(t)} = F2(s) When does {f1(t) + f2(t)} = F1(s) + F2(s)?Figure 7.1.4 suggests, but does not prove, that the function f(t)=et2is not of exponential order. How does the observation that t2 ln M + ct, for M 0 and t sufficiently large, show that et2Mect for any c?Discussion Problems Use part (c) of Theorem 7.1.1 to show that e(a+ib)t=sa+ib(sa)2+b2, where a and b are real and i2 = 1. Show how Eulers formula (page 136) can then be used to deduce the results eatcosbt=sa(sa)2+b2eatsinbt=b(sa)2+b2.50E51EShow that the function f(t)=1/t2 does not possess a Laplace transform. [Hint Write L{1/t2} as two improper integrals: L{1/t2}=01estt2dt+1estt2dt=I1+I2. Show that I1 diverges.]53E54E55E56E57EIn Problems 5558 use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that a and k are positive constants. 58. L{sintsinht}=2ss4+4;L{sinktsinhkt}In Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. 1. 1{1s3}2E3E4EIn Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. 5. 1{(s+1)3s4}In Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. 6. 1{(s+2)2s3}7E8E9EIn Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. 10. 1{15s2}In Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. 11. 1{5s2+49}12E13E14E15EIn Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. 16. 1{s+1s2+2}In Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. L1{1s2+3s}In Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. L1{s+1s24s}In Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. L1{ss2+2s3}In Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. L1{1s2+s20}In Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. L1{0.9s(s0.1)(s+0.2)}22EIn Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. L1{s(s2)(s3)(s6)}24E25E26E27E28EIn Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. 29. L1{1(s2+1)(s2+4)}In Problems 130 use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform. 30. L1{6s+3s4+5s2+4}In Problems 3134 find the given inverse Laplace transform by finding the Laplace transform of the indicated function f. 31. 1{1(sa)2b2};f(t)=eatsinhbtIn Problems 3134 find the given inverse Laplace transform by finding the Laplace transform of the indicated function f. 32. 1{1s2(s2+a2)};f(t)=atsinatIn Problems 3134 find the given inverse Laplace transform by finding the Laplace transform of the indicated function f. 33. 1{1(s2+a2)(s2+b2)};f(t)=asinbtbsinatIn Problems 3134 find the given inverse Laplace transform by finding the Laplace transform of the indicated function f. 34. 1{s(s2+a2)(s2+b2)};f(t)=cosbtcosatIn Problems 3544 use the Laplace transform to solve the given initial-value problem. 35. dydty=1, y(0) = 0In Problems 3544 use the Laplace transform to solve the given initial-value problem. 36. 2dydt+y=0, y(0) = 3In Problems 3544 use the Laplace transform to solve the given initial-value problem. 37. y + 6y = e4t, y(0) = 2In Problems 3544 use the Laplace transform to solve the given initial-value problem. 38. y y = 2 cos 5t, y(0) = 0In Problems 3544 use the Laplace transform to solve the given initial-value problem. 39. y + 5y + 4y = 0, y(0) = 1, y(0) = 0In Problems 3544 use the Laplace transform to solve the given initial-value problem. 40. y 4y = 6e3t 3et, y(0) = 1, y(0) = 1In Problems 3544 use the Laplace transform to solve the given initial-value problem. 41. y+y=2sin2t, y(0) = 10, y(0) = 042EIn Problems 3544 use the Laplace transform to solve the given initial-value problem. 43. 2y + 3y 3y 2y = et, y(0) = 0, y(0) = 0, y(0) = 1In Problems 3544 use the Laplace transform to solve the given initial-value problem. 44. y + 2y y 2y = sin 3t, y(0) = 0, y(0) = 0, y(0) = 1The inverse forms of the results in Problem 49 in Exercises 7.1 are 1{sa(sa)2+b2}=eatcosbt1{b(sa)2+b2}=eatsinbt. y + y = e3t cos 2t, y(0) = 046E47E