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Bartleby Sitemap - Textbook Solutions

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

10E 11E 12E In Problems 118 solve the given differential equation. 13. 3x2y + 6xy + y = 0 In Problems 118 solve the given differential equation. 14. x2y 7xy + 41y = 0 15E 16E In Problems 118 solve the given differential equation. 17. xy(4) + 6y = 0 18E In Problems 19-24 solve the given partial equation by variation of parameters. 19. xy 4y = x4 20E 21E 22E In Problems 19-24 solve the given partial equation by variation of parameters. 23. x2y + xy y = ln x In Problems 1924 solve the given differential equation by variation of parameters. 24. x2y+xyy=1x+1 In Problems 2530 solve the given initial-value problem. Use a graphing utility to graph the solution curve. 25. x2y+3xy=0,y(1)=0,y(1)=4 26E 27E In Problems 25-30 solve the given initial-value problem. Use a graphing utility to graph the solution curve. 28. x2y 3xy + 4y = 0, y(1) = 56, y(1) = 3 29E In Problems 25-30 solve the given initial-value problem. Use a graphing utility to graph the solution curve. 30. x2y 5xy + 8y = 8x6, y12=0,y12=0 In Problems 3136 use the substitution x = et to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 4.34.5. 31. x2y + 9xy 20y = 0 32E In Problems 3136 use the substitution x = et to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 4.34.5. 33. x2y + 10xy + 8y = x2 In Problems 3136 use the substitution x = et to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 4.34.5. 34. x2y 4xy + 6y = ln x2 35E In Problems 3136 use the substitution x = et to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 4.34.5. 36. x3y 3x2y + 6xy 6y = 3 + ln x3 37E In Problems 37 and 38 use the substitution t = x to solve the given initial-value problem on the interval (, 0). 38. x2y 4xy + 6y = 0, y(2) = 8, y (2) = 0 39E 40E 41E 42E 43E 44E 45E 46E Bending of a Circular Plate In the analysis of the bending of a uniformly loaded circular plate, the equation w(r) of the deflection curve of the plate can be shown to satisfy the differential equation d2wdr3+1rd2wdr21r2dwdr=q2Dr,(9) where q and D are constants. Here r is the radial distance from a point on the circular plate to its center. (a) Use the method of this section along with variation of parameters as given in (15) of Section 4.6 to find the general solution of equation (9). (b) Find a solution of (9) that satisfies the boundary conditions w(0)=0,w(a)=0,w(a)=0, where a 0 is the radius of the plate. [Hint: The condition w(0) = 0 is correct. Use this condition to determine one of the constants in the general solution found in part (a).]48E In Problems 16 proceed as in Example 1 to find a particular solution yp(x) of the given differential equation in the integral form (10). 1. y 16y = f(x)In Problems 16 proceed as in Example 1 to find a particular solution yp(x) of the given differential equation in the integral form (10). 2. y + 3y 10y = f(x)3E 4E 5E 6E 7E In Problems 712 proceed as in Example 2 to find the general solution of the given differential equation. Use the results obtained in Problems 16. Do not evaluate the integral that defines yp(x). 8. y + 3y 10y = x2 9E 10E 11E 12E 13E 14E 15E 16E 17E In Problems 13-18 proceed as in Example 3 to find a solution of the given initial-value problem. Evaluate the integral that defines yp(x). 18. y + y = sec2 x cot x, y() = 0, y() = 0 19E 20E 21E 22E In Problems 19-30 proceed as in Example 5 to find a solution of the given initial-value problem. 23. y + y = csc x cot x, y(/2) = /2, y(/2) = 1 24E 25E 26E 27E 28E 29E 30E 31E In Problems 31-34 proceed as in Example 6 to find a solution of the initial-value problem with the given piecewise-defined forcing function. 32. y y = f(x), y(0) = 3, y(0) = 2, where f(x)={0,x0x,x0 33E 34E 35E 36E 37E 38E 39E 40E 41E In Problems 39-44 proceed as in Examples 7 and 8 to find a solution of the given boundary-value problem. 42. y y = e2x, y(0) = 0, y(1) = 0 43E 44E In Problems 120 solve the given system of differential equations by systematic elimination. 1. dxdt=2xydydt=x 2E 3E 4E 5E 6E 7E In Problems 120 solve the given system of differential equations by systematic elimination. 8. d2xdt2+dydt=5x 9E 10E 11E 12E 13E 14E 15E 16E 17E In Problems 120 solve the given system of differential equations by systematic elimination. 18. Dx+z=et(D1)x+Dy+Dz=0x+2y+Dz=et 19E 20E 21E 22E Projectile Motion A projectile shot from a gun has weight w = mg and velocity v tangent to its path of motion. Ignoring air resistance and all other forces acting on the projectile except its weight, determine a system of differential equations that describes its path of motion. See Figure 4.9.2. Solve the system. [Hint. Use Newtons second law of motion in the x and y directions.] FIGURE 4.9.2 Path of projectile in Problem 23 24E 25E 1E 2E In Problems 3-6 the dependent variable y is missing in the given differential equation. Proceed as in Example 1 and solve the equation by using the substitution u = y. 3. y + (y)2 + 1 = 0 4E In Problems 3-6 the dependent variable y is missing in the given differential equation. Proceed as in Example 1 and solve the equation by using the substitution u = y. 5. x2y + (y)2 = 0 6E In Problems 710 the independent variable x is missing in the given differential equation. Proceed as in Example 2 and solve the equation by using the substitution u = y. 7. yy + (y)2 + 1 = 0 8E 9E 10E 11E In Problems 11 and 12 solve the given initial-value problem. 12. y + x(y)2 = 0, y(1) = 4, y(1) = 2 13E 14E 15E In Problems 15 and 16 show that the substitution u = y leads to a Bernoulli equation. Solve this equation (see Section 2.5). 16. xy = y + x(y)2 In Problems 17–20 proceed as in Example 3 and obtain the first six nonzero terms of a Taylor series solution, centered at 0, of the given initial-value problem. Use a numerical solver and a graphing utility to compare the solution curve with the graph of the Taylor polynomial. 17. y″ = x + y2, y(0) = 1, y′(0) = 1 18E 19E 20E In calculus the curvature of a curve that is defined by a function y = f(x) is defined as =y[1+(y)2]3/2 Find y = f(x) for which = 1. [Hint. For simplicity, ignore constants of integration.22E Discuss how the method of reduction of order considered in this section can be applied to the third-order differential equation y=1+(y)2. Carry out your ideas and solve the equation.24E 25E Answer Problems 110 without referring back to the text. Fill in the blank or answer true or false. 1. The only solution of the initial-value problem y + x2y = 0, y(0) = 0, y(0) = 0 is __________.Answer Problems 110 without referring back to the text. Fill in the blank or answer true or false. 2. For the method of undetermined coefficients, the assumed form of the particular solution yp for y y = 1 + ex is __________.Answer Problems 110 without referring back to the text. Fill in the blank or answer true or false. 3. A constant multiple of a solution of a linear differential equation is also a solution. __________4RE 5RE 6RE 7RE 8RE 9RE If y1 = ex and y2 = ex are solutions of homogeneous linear differential equation, then necessarily y = 5ex + 10ex is also a solution of the DE. ______________11RE Without the aid of the Wronskian, determine whether the given set of functions is linearly independent or linearly dependent on the indicated interval. (a) f1(x) = ln x, f2(x) = ln x2, (0, ) (b) f1(x) = xn, f2(x) = xn+1, n = 1, 2, , (, ) (c) f1(x) = x, f2(x) = x + 1, (, ) (d) f1(x)=cos(x+2),f2(x)=sinx,(,) (e) f1(x) = 0, f2(x) = x, (5, 5) (f) f1(x) = 2, f2(x) = 2x, (, ) (g) f1(x) = x2, f2(x) = 1 x2, f3(x) = 2 + x2, (, ) (h) f1(x)=xex+1,f2(x)=(4x5)ex,f3(x)=xex,(,)Suppose m1 = 3, m2 = 5, and m3 = 1 are roots of multiplicity one, two, and three, respectively, of an auxiliary equation. Write down the general solution of the corresponding homogeneous linear DE if it is (a) an equation with constant coefficients, (b) a Cauchy-Euler equation.14RE In Problems 15 and 16 find a homogeneous second-order Cauchy-Euler equation with real coefficients if the given numbers are roots of its auxiliary equation. 15. m1 = 4, m2 = 1 16RE In Problems 1732 use the procedures developed in this chapter to find the general solution of each differential equation. 17. y 2y 2y = 0 18RE 19RE 20RE In Problems 1732 use the procedures developed in this chapter to find the general solution of each differential equation. 21. 3y + 10y + 15y + 4y = 0 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE 30RE 31RE 32RE In Problems 33 and 34 write down the form of the general solution y = yc + yp of the given differential equation in the two cases and = . Do not determine the coefficients in yp. 33. y + 2y = sin x 34RE 35RE 36RE 37RE 38RE 39RE 40RE 41RE 42RE 43RE 44RE 46RE 47RE In Problems 47-50 use systematic elimination to solve the given system. 48. dxdt=2x+y+t2dydt=3x+4y4t 49RE 50RE 5.1.1 Spring/Mass systems: Free Undamped Motion A mass weighing 4 pounds is attached to a spring whose spring constant is 16 lb/ft. What is the period of simple harmonic motion?Spring/Mass Systems: Free Undamped Motion A 20-kilogram mass is attached to a spring. If the frequency of simple harmonic motion is 2/ cycles/s, what is the spring constant k? What is the frequency of simple harmonic motion if the original mass is replaced with an 80-kilogram mass?Spring/Mass Systems: Free Undamped Motion A mass weighing 24 pounds, attached to the end of a spring, stretches it 4 inches. Initially, the mass is released from rest from a point 3 inches above the equilibrium position. Find the equation of motion.Spring/Mass Systems: Free Undamped Motion Determine the equation of motion if the mass in Problem 3 is initially released from the equilibrium position with a downward velocity of 2 ft/s.Spring/Mass Systems: Free Undamped Motion A mass weighing 20 pounds stretches a spring 6 inches. The mass is initially released from rest from a point 6 inches below the equilibrium position. (a) Find the position of the mass at the times t = /12, /8, /6, /4, and 9/32 s. (b) What is the velocity of the mass when t = 3/16 s? In which direction is the mass heading at this instant? (c) At what times does the mass pass through the equilibrium position?Spring/Mass Systems: Free Undamped Motion A force of 400 newtons stretches a spring 2 meters. A mass of 50 kilograms is attached to the end of the spring and is initially released from the equilibrium position with an upward velocity of 10 m/s. Find the equation of motion.7E Spring/Mass Systems: Free Undamped Motion A mass weighing 32 pounds stretches a spring 2 feet. Determine the amplitude and period of motion if the mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of 2 ft/s. How many complete cycles will the mass have completed at the end of 4 seconds?9E 5.1.1Spring/Mass Systems: Free Undamped Motion A mass weighing 10 pounds stretches a spring 14 foot. This mass is removed and replaced with a mass of 1.6 slugs, which is initially released from a point 13 foot above the equilibrium position with a downward velocity of 54 ft/s. (a) Express the equation of motion in the form given in (6). (b) Express the equation of motion in the form given in (6) (c) Use one of the solutions obtained in parts (a) and (b) to determine the times the mass attains a displacement below the equilibrium position numerically equal to 12 the amplitude of motion.A mass weighing 64 pounds stretches a spring 0.32 foot. The mass is initially released from a point 8 inches above the equilibrium position with a downward velocity of 5 ft/s. (a) Find the equation of motion. (b) What are the amplitude and period of motion? (c) How many complete cycles will the mass have completed at the end of 3 seconds? (d) At what time does the mass pass through the equilibrium position heading downward for the second time? (e) At what times does the mass attain its extreme displacements on either side of the equilibrium position? (f) What is the position of the mass at t = 3 s? (g) What is the instantaneous velocity at t = 3 s? (h) What is the acceleration at t = 3 s? (i) What is the instantaneous velocity at the times when the mass passes through the equilibrium position? (j) At what times is the mass 5 inches below the equilibrium position? (k) At what times is the mass 5 inches below the equilibrium position heading in the upward direction?A mass of 1 slug is suspended from a spring whose spring constant is 9 lb/ft. The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of 3 ft/s. Find the times at which the mass is heading downward at a velocity of 3 ft/s.13E 14E 15E 16E 17E 18E Spring/Mass Systems: Free Undamped Motion A model of a spring/mass system is 4x+e0.1tx=0. By inspection of the differential equation only, discuss the behavior of the system over a long period of time.20E 5.1.2 Spring/Mass systems: Free Damped Motion In Problems 2124 the given figure represents the graph of an equation of motion for a damped spring/mass system. Use the graph to determine (a) whether the initial displacement is above or below the equilibrium position and (b) whether the mass is initially released from rest, heading downward, or heading upward. 21. Figure 5.1.18 Graph for Problem 21 Spring/Mass Systems: Free Damped Motion In Problems 2124 the given figure represents the graph of an equation of motion for a damped spring/mass system. Use the graph to determine (a) whether the initial displacement is above or below the equilibrium position and (b) whether the mass is initially released from rest, heading downward, or heading upward. Figure 5.1.18 Graph for Problem 21 Spring/Mass Systems: Free Damped Motion In Problems 2124 the given figure represents the graph of an equation of motion for a damped spring/mass system. Use the graph to determine (a) whether the initial displacement is above or below the equilibrium position and (b) whether the mass is initially released from rest, heading downward, or heading upward. Figure 5.1.19 Graph for Problem 22 24E Spring/Mass System: Free Damped Motion A mass weighing 4 pounds is attached to a spring whose constant is 2 lb/ft. The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from a point 1 foot above the equilibrium position with a downward velocity of 8 ft/s. Determine the time at which the mass passes through the equilibrium position. Find the time at which the mass attains its extreme displacement from the equilibrium position. What is the position of the mass at this instant?Spring/Mass Systems: Free Damped Motion A 4-foot spring measures 8 feet long after a mass weighing 8 pounds is attached to it. The medium through which the mass moves offers a damping force numerically equal to 2 times the instantaneous velocity. Find the equation of motion if the mass is initially released from the equilibrium position with a downward velocity of 5 ft/s. Find the time at which the mass attains its extreme displacement from the equilibrium position. What is the position of the mass at this instant?A 1-kilogram mass is attached to a spring whose constant is 16 N/m, and the entire system is then submerged in a liquid that imparts a damping force numerically equal to 10 times the instantaneous velocity. Determine the equations of motion if (a) the mass is initially released from rest from a point 1 meter below the equilibrium position, and then (b) the mass is initially released from a point 1 meter below the equilibrium position with an upward velocity of 12 m/s.28E Spring/Mass Systems: Free Damped Motion A force of 2 pounds stretches a spring 1 foot. A mass weighing 3.2 pounds is attached to the spring, and the system is then immersed in a medium that offers a damping force that is numerically equal to 0.4 times the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from rest from a point 1 foot above the equilibrium position. (b) Express the equation of motion in the form given in (23). (c) Find the first time at which the mass passes through the equilibrium position heading upward.After a mass weighing 10 pounds is attached to a 5-foot spring, the spring measures 7 feet. This mass is removed and replaced with another mass that weighs 8 pounds. The entire system is placed in a medium that offers a damping force that is numerically equal to the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from a point 12 foot below the equilibrium position with a downward velocity of 1 ft/s. (b) Express the equation of motion in the form given in (23). (c) Find the times at which the mass passes through the equilibrium position heading downward. (d) Graph the equation of motion.Spring/Mass Systems: Free Damped Motion A mass weighing 10 pounds stretches a spring 2 feet. The mass is attached to a dashpot device that offers a damping force numerically equal to ( 0) times the instantaneous velocity. Determine the values of the damping constant so that the subsequent motion is (a) overdamped, (b) critically damped, and (c) underdamped.f(t)=cos5t+sin2t Spring/Mass Systems: Free Damped Motion A mass weighing 16 pounds stretches a spring 83 feet. The mass is initially released from rest from a point 2 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 12 the instantaneous velocity. Find the equation of motion if the mass is driven by an external force equal to f (t) = 10 cos 3t.A mass of 1 slug is attached to a spring whose constant is 5 lb/ft. Initially, the mass is released 1 foot below the equilibriumposition with a downward velocity of 5 ft/s, and the subsequentmotion takes place in a medium that offers a damping force thatis numerically equal to 2 times the instantaneous velocity. (a) Find the equation of motion if the mass is driven by anexternal force equal to f(t) = 12 cos 2t + 3 sin 2t. (b) Graph the transient and steady-state solutions on the samecoordinate axes. (c) Graph the equation of motion.35E In Problem 35 determine the equation of motion if the external force is f(t) = et sin 4t. Analyze the displacements for t .Spring/Mass Systems: Driven Motion When a mass of 2 kilograms is attached to a spring whose constant is 32 N/m, it comes to rest in the equilibrium position. Starting at t = 0. a force equal to f(t)=68e2tcos4t is applied to the system. Find the equation of motion in the absence of damping.Spring/Mass Systems: Driven Motion In Problem 37 write the equation of motion in the form x(t)=Asin(t+)+Be2tsin(4t+). What is the amplitude of vibrations after a very long time?Spring/Mass Systems: Driven Motion A mass m is attached to the end of a spring whose constant is k. After the mass reaches equilibrium, its support begins to oscillate vertically about a horizontal line L according to a formula h(t). The value of h represents the distance in feet measured from L. See Figure 5.1.22. (a) Determine the differential equation of motion if the entire system moves through a medium offering a damping force that is numerically equal to (dx/dt). (b) Solve the differential equation in part (a) if the spring is stretched 4 feet by a mass weighing 16 pounds and = 2, h(t) = 5 cos t, x(0) = x(0) = 0. FIGURE 5.1.22 Oscillating support in Problem 39 A mass of 100 grams is attached to a spring whose constant is 1600 dynes/cm. After the mass reaches equilibrium, its supportoscillates according to the formula h(t) = sin 8t, where hrepresents displacement from its original position. SeeProblem 39 and Figure 5.1.22. (a) In the absence of damping, determine the equation ofmotion if the mass starts from rest from the equilibriumposition. (b) At what times does the mass pass through the equilibriumposition? (c) At what times does the mass attain its extreme displacements? (d) What are the maximum and minimum displacements? (e) Graph the equation of motion. 39. A mass m is attached to the end of a spring whose constant is k. After the mass reaches equilibrium, its support beginsto oscillate vertically about a horizontal line L according toa formula h(t). The value of h represents the distance in feetmeasured from L. See Figure 5.1.22. (a) Determine the differential equation of motion if the entiresystem moves through a medium offering a damping forcethat is numerically equal to (dx/dt). (b) Solve the differential equation in part (a) if the spring isstretched 4 feet by a mass weighing 16 pounds and = 2,h(t) = 5 cos t, x(0) = x'(0) = 0. FIGURE 5.1.22 Oscillating support in Problem 39 Spring/Mass Systems: Driven Motion In Problems 41 and 42 solve the given initial-value problem. 41. d2xdt2+4x=5sin2t+3cos2t,x(0)=1,x(0)=1 In Problems 41 and 42 solve the given initial-value problem. 42. d2xdt2+9x=5sin3t, x(0) = 2, x(0) = 0 Series Circuit Analogue (a) Show that the solution of the initial-value problem d2xdt2+2x=F0cost,x(0)=0,x(0)=0 is x(t)F022(costcost). (b) Evaluate limF022(costcost).Compare the result obtained in part (b) of Problem 43 with the solution obtained using variation of parameters when the external force is F0 cos t. 43. (a) Show that the solution of the initial-value problem d2xdt2+2x=F0cost,x(0)=0,x(0)=0 is x(t)=F022(costcost). (b) Evaluate limF022(costcost).(a) Show that x(t) given in part (a) of Problem 43 can be written in the form x(t)=2F022sin12()tsin12(+)t (b) If we define =12(), show that when is small an approximate solution is x(t)=F02sintsint. When is small, the frequency /2 of the impressed force is close to the frequency /2 of free vibrations. When this occurs, the motion is as indicated in Figure 5.1.23. Oscillations of this kind are called beats and are due to the fact that the frequency of sin t is quite small in comparison to the frequency of sin t. The dashed curves, or envelope of the graph of x(t), are obtained from the graphs of (F0/2) sin t. Use a graphing utility with various values of F0, , and to verify the graph in Figure 5.1.23. Figure 5.1.23 Beats phenomenon in Problem 45 43. (a) Show that the solution of the initial-value problem d2xdt2+2x=F0cost,x(0)=0,x(0)=0 is x(t)=F022(costcost).Series Circuit Analogue Find the charge on the capacitor in an LRC-series circuit at t = 0.01 s when L = 0.05 h, R = 2 , C = 0.01 f, E(t) = 0 V, q(0) = 5 C, and i(0) = 0 A. Determine the first time at which the charge on the capacitor is equal to zero.Series Circuit Analogue Find the charge on the capacitor in an LRC-series circuit when L=14h, R=20, C=1300f, E(t)=0V, q(0)=4C, and i(0)=0A. Is the charge on the capacitor ever equal to zero?Series Circuit Analogue In Problems 51 and 52 find the charge on the capacitor and the current in the given LRC-series circuit. Find the maximum charge on the capacitor. 51. L=53h, R = 10 , C=130f, E(t) = 300 V, q(0) = 0 C, i(0) = 0 A In Problems 51 and 52 find the charge on the capacitor and the current in the given LRC-series circuit. Find the maximum charge on the capacitor. 52. L = 1 h, R = 100 , C = 0.0004 f, E(t) = 30 V, q(0) = 0 C, i(0) = 2 A 53E 54E 55E Find the steady-state current in an LRC-series circuit when L=12h, R = 20 , C = 0.001 f, and E(t) = 100 sin 60t + 200 cos 40t V.57E 58E 59E Series Circuit Analogue Find the charge on the capacitor and the current in an LC-series circuit when L = 0.1 h, C = 0.1 f, E(t) = 100 sin t V, q(0) = 0 C, and i(0) = 0 A.61E 62E (a) The beam is embedded at its left end and free at its right end, and w(x) = w0, 0 x L. (b) Use a graphing utility to graph the deflection curve when w0 = 24EI and L = 1.(a) The beam is simply supported at both ends, and w(x) = w0, 0 x L. (b) Use a graphing utility to graph the deflection curve w0 = 24EI and L = 1.(a) The beam is embedded at its left end and simply supported at its right end, and w(x) = w0, 0 x L. (b) Use a graphing utility to graph the deflection curve when w0 = 48EI and L = 1.(a) The beam is embedded at its left end and simply supported at its right end, and w(x) = w0 sin(x/L), 0 x L. (b) Use a graphing utility to graph the deflection curve when w0 = 23EI and L = 1.6E A cantilever beam of length L is embedded at its right end, and a horizontal tensile force of P pounds is applied to its free left end. When the origin is taken at its free end, as shown in Figure 5.2.8, the deflection y(x) of the beam can be shown to satisfy the differential equation EIy=Pyw(x)x2. Find the deflection of the cantilever beam if w(x) = w0x, 0 x L, and y(0) = 0, y(L) = 0. Figure 5.2.8 Deflection of cantilever beam in Problem 7 8E 9E In Problems 920 find the eigenvalues and eigenfunctions for the given boundary-value problem. 10. y + y = 0, y(0) = 0, y(/4) = 0 In Problems 920 find the eigenvalues and eigenfunctions for the given boundary-value problem. 11. y + y = 0, y(0) = 0, y(L) = 0 In Problems 920 find the eigenvalues and eigenfunctions for the given boundary-value problem. 12. y + y = 0, y(0) = 0, y(/2) = 0 13E In Problems 920 find the eigenvalues and eigenfunctions for the given boundary-value problem. 14. y + y = 0, y() = 0, y() = 0 In Problems 920 find the eigenvalues and eigenfunctions for the given boundary-value problem. 15. y + 2y + ( + 1)y = 0, y(0) = 0, y(5) = 0 16E 17E 18E Eigenvalues and Eigenfunctions In Problems 920 find the eigenvalues and eigenfunctions for the given boundary-value problem. 19. x2y + xy + y = 0, y(1) = 0, y(e2) = 0 Eigenvalues and Eigenfunctions In Problems 920 find the eigenvalues and eigenfunctions for the given boundary-value problem. 20. x2y + xy + y = 0, y(1) = 0, y(e) = 0 21E 22E 23E The critical loads of thin columns depend on the end conditions of the column. The value of the Euler load P1 in Example 4 was derived under the assumption that the column was hinged at both ends. Suppose that a thin vertical homogeneous column is embedded at its base (x = 0) and free at its top (x = L) and that a constant axial load P is applied to its free end. This load either causes a small deflection as shown in Figure 5.2.9 or does not cause such a deflection. In either case the differential equation for the deflection y(x) is EId2ydx+Py=P Figure 5.2.9 Deflection of vertical column in Problem 24 (a) What is the predicted deflection when = 0? (b) When 0, show that the Euler load for this column is one-fourth of the Euler load for the hinged column in Example 4.25E Rotating String Consider the boundary-value problem introduced in the construction of the mathematical model for the shape of a rotating string: Td2ydx2+2y=0,y(0)=0,y(L)=0. For constant T and , define the critical speeds of angular rotation n as the values of for which the boundary-value problem has nontrivial solutions. Find the critical speeds n and the corresponding deflections yn(x).28E Additional Boundary-Value Problems Temperature in a Sphere Consider two concentric spheres of radius r = a and r = b, a b. See Figure 5.2.10. The temperature u(r) in the region between the spheres is determined from the boundary-value problem rd2udr2+2dudr=0,u(a)=u0,u(b)=u1, where u0 and u1 are constants. Solve for u(r). FIGURE 5.2.10 Concentric spheres in Problem 29 30E 31E 32E 33E Damped Motion Assume that the model for the spring/mass system in Problem 33 is replaced by mx + 2x + kx = 0. In other words, the system is free but is subjected to damping numerically equal to 2 times the instantaneous velocity. With the same initial conditions and spring constant as in Problem 33, investigate whether a mass m can be found that will pass through the equilibrium position at t = 1 second.Additional Boundary-Value Problems y + 16y = 0, y(0) = y0, y(/2) = y1 Additional Boundary-Value Problems y + 16y = 0, y(0) = 1, y(L) = 1 Consider the boundary-value problem y+y=0,y()=y(),y()=y(). (a) The type of boundary conditions specified are called periodic boundary conditions. Give a geometric interpretation of these conditions. (b) Find the eigenvalues and eigenfunctions of the problem. (c) Use a graphing utility to graph some of the eigenfunctions. Verify your geometric interpretation of the boundary conditions given in part (a).Show that the eigenvalues and eigenfunctions of the boundary-value problem y + y = 0, y(0) = 0, y(1) + y(1) = 0 are n = n2 and yn(x) = sin nx, respectively, where n, n = 1, 2, 3, are the consecutive positive roots of the equation tan = .Find a linearization of the differential equation in Problem 4. d2xdt2+xe0.01x=0,14E 15E A uniform chain of length L, measured in feet, is held vertically so that the lower end just touches the floor. The chain weighs 2 lb/ft. The upper end that is held is released from rest at t = 0 and the chain falls straight down. If x(t) denotes the length of the chain on the floor at time t, air resistance is ignored, and the positive direction is taken to be downward, then (Lx)d2xdt2(dxdt)2=Lg. (a) Solve for v in terms of x. Solve for x in terms of t. Express v in terms of t. (b) Determine how long it takes for the chain to fall completely to the ground. (c) What velocity does the model in part (a) predict for the upper end of the chain as it hits the ground?Pursuit curve In a naval exercise a ship S1 is pursued by a submarine S2 as shown in Figure 5.3.9. Ship S1 departs point (0, 0) at t = 0 and proceeds along a straight-line course (the y-axis) at a constant speed v1. The submarine S2 keeps ship S1 in visual contact, indicated by the straight dashed line L in the figure, while traveling at a constant speed v2 along a curve C. Assume that ship S2 starts at the point (a, 0), a 0, at t = 0 and that L is tangent to C. (a) Determine a mathematical model that describes the curve C. (b) Find an explicit solution of the differential equation. For convenience define r = v1/v2. (c) Determine whether the paths of S1 and S2 will ever intersect by considering the cases r 1, r 1, and r = 1. [Hint: dtdx=dtdsdsdx, where s is arc length measured along C.] Figure 5.3.9 Pursuit curve in Problem 17 Pursuit curve In another naval exercise a destroyer S1 pursues a submerged submarine S2. Suppose that S1 at (9, 0) on the x-axis detects S2 at (0, 0) and that S2 simultaneously detects S1. The captain of the destroyer S1 assumes that the submarine will take immediate evasive action and conjectures that its likely new course is the straight line indicated in Figure 5.3.10. When S1 is at (3, 0), it changes from its straight-line course toward the origin to a pursuit curve C. Assume that the speed of the destroyer is, at all times, a constant 30 mi/h and that the submarines speed is a constant 15 mi/h. (a) Explain why the captain waits until S1 reaches (3, 0) before ordering a course change to C. (b) Using polar coordinates, find an equation r = f () for the curve C. (c) Let T denote the time, measured from the initial detection, at which the destroyer intercepts the submarine. Find an upper bound for T.19E 21E If a mass weighing 10 pounds stretches a spring 2.5 feet, a mass weighing 32 pounds will stretch it __________ feet.2RE 3RE Pure resonance cannot take place in the presence of a damping force. __________5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE

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