Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus Volume 3
Find the general solution to the linear differential equation. 16. 3y14y+8y=0Find the general solution to the linear differential equation. 17. y+y+y=0Find the general solution to the linear differential equation. 18. 5y+2y+4y=0Find the general solution to the linear differential equation. 19. y121y=0Find the general solution to the linear differential equation. 20. 8y+14y15y=0Find the general solution to the linear differential equation. 21. y+81y=0Find the general solution to the linear differential equation. 22. yy+11y=0Find the general solution to the linear differential equation. 23. 2y=0Find the general solution to the linear differential equation. 24. y6y+9y=0Find the general solution to the linear differential equation. 25. 3y2y7y=0Find the general solution to the linear differential equation. 26. 4y10y=0Find the general solution to the linear differential equation. 27. 36d2ydx2+12dydx+y=0Find the general solution to the linear differential equation. 28. 25d2ydx280dydx+64y=0Find the general solution to the linear differential equation. 29. d2ydx29dydx=0Find the general solution to the linear differential equation. 30. 4d2ydx2+8y=0Solve the initial-value problem. 31. y+5y+6y=0 , y(0)=0 , y(0)=2Solve the initial-value problem. 32. y+2y8y=0 , y(0)=5 , y(0)=4Solve the initial-value problem. 33. y+4y=0 , y(0)=3 , y(0)=10Solve the initial-value problem. 34. y18y+81y=0 , y(0)=1 , y(0)=5Solve the initial-value problem. 35. yy30y=0 , y(0)=1 , y(0)=16Solve the initial-value problem. 36. 4y+4y8y=0 , y(0)=2 , y(0)=1Solve the initial-value problem. 37. 25y+10y+y=0 , y(0)=2 , y(0)=1Solve the initial-value problem. 38. y+y=0 , y()=1 , y()=5Solve the boundary-value problem, if possible. 39. y+y42y=0 , y(0)=0 , y(1)=2Solve the boundary-value problem, if possible. 40. 9y+y=0 , y(32)=6 , y(0)=8Solve the boundary-value problem, if possible. 41. y+10y+34y=0 , y(0)=6 , y()=2Solve the boundary-value problem, if possible. 42. y+7y60y=0 , y(0)=4 , y(2)=0Solve the boundary-value problem, if possible. 43. y4y+4y=0 , y(0)=2 , y(1)=1Solve the boundary-value problem, if possible. 44. y5y=0 , y(0)=3 , y(1)=2Solve the boundary-value problem, if possible. 45. y+9y=0 , y(0)=4 , y(3)=4Solve the boundary-value problem, if possible. 46. 4y+25y=0 , y(0)=2 , y(2)=2Find a differential equation with a general solution that is y=c1ex/5+c2e4x .Find a differential equation with a general solution that is y=c1ex+c2e4x/3 .For each of the following differential equations: a. Solve the initial value problem. b. [T] Use a graphing utility to graph the particular solution. 49. y+64y=0 ; y(0)=3 , y(0)=16For each of the following differential equations: a. Solve the initial value problem. b. [T] Use a graphing utility to graph the particular solution. 50. y2y+10y=0 y(0)=1 , y(0)=13For each of the following differential equations: a. Solve the initial value problem. b. [T] Use a graphing utility to graph the particular solution. 51. y+5y+15y=0 y(0)=2 , y(0)=7(Principle of superposition) Prove that if y1(x) and y2(x) are solutions to a linear homogeneous differential equation, y+p(x)y+q(x)y=0 , then the function y(x)=c1y1(x)+c2y2(x) , where c1 and c2 are constants, is also a solution.Prove that if a, b, and c are positive constants, then all solutions to the second-order linear differential equation ay+by+cy=0 approach zero as x (Hint: Consider three cases: two distinct roots, repeated real roots, and complex conjugate roots.)Solve the following equations using the method of undetermined coefficients. 54. 2y5y12y=6Solve the following equations using the method of undetermined coefficients. 55. 3y+y4y=8Solve the following equations using the method of undetermined coefficients. 56. y6y+5y=exSolve the following equations using the method of undetermined coefficients. 57. y+16y=e2xSolve the following equations using the method of undetermined coefficients. 58. y4y=x2+1Solve the following equations using the method of undetermined coefficients. 59. y4y+4y=8x2+4xSolve the following equations using the method of undetermined coefficients. 60. y2y3y=sin2xSolve the following equations using the method of undetermined coefficients. 61. y+2y+y=sinx+cosxSolve the following equations using the method of undetermined coefficients. 62. y+9y=excosxSolve the following equations using the method of undetermined coefficients. 63. y+y=3sin2x+xcos2xSolve the following equations using the method of undetermined coefficients. 64. y+3y28y=10e4xSolve the following equations using the method of undetermined coefficients. 65. y+10y+25y=xe5x+4In each of the following problems, a. Write the form for the particular solution yp(x) for the method of undetermined coefficients. b. [T] Use a computer algebra system to find a particular solution to the given equation. 66. yyy=x+exIn each of the following problems, a. Write the form for the particular solution yp(x) for the method of undetermined coefficients. b. [T] Use a computer algebra system to find a particular solution to the given equation. 67. y3y=x24x+11In each of the following problems, a. Write the form for the particular solution yp(x) for the method of undetermined coefficients. b. [T] Use a computer algebra system to find a particular solution to the given equation. 68. yy4y=excos3xIn each of the following problems, a. Write the form for the particular solution yp(x) for the method of undetermined coefficients. b. [T] Use a computer algebra system to find a particular solution to the given equation. 69. 2yy+y=(x25x)exIn each of the following problems, a. Write the form for the particular solution yp(x) for the method of undetermined coefficients. b. [T] Use a computer algebra system to find a particular solution to the given equation. 70. 4y+5y2y=e2x+xsinxIn each of the following problems, a. Write the form for the particular solution yp(x) for the method of undetermined coefficients. b. [T] Use a computer algebra system to find a particular solution to the given equation. 71. yy2y=x2exsinxSolve the differential equation using either the method of undetermined coefficients or the variation of parameters. 72. y+3y4y=2exSolve the differential equation using either the method of undetermined coefficients or the variation of parameters. 73. y+2y=e3xSolve the differential equation using either the method of undetermined coefficients or the variation of parameters. 74. y+6y+9y=exSolve the differential equation using either the method of undetermined coefficients or the variation of parameters. 75. y+2y8y=6e2xSolve the differential equation using the method of variation of parameters. 76. 4y+y=2sinxSolve the differential equation using the method of variation of parameters. 77. y9y=8xSolve the differential equation using the method of variation of parameters. 78. y+y=secx , 0x/2Solve the differential equation using the method of variation of parameters. 79. y+4y=3csc2x , 0x/2Find the unique solution satisfying the differential equation and the initial conditions given, where yp(x) is the particular solution. 80. y2y+y=12ex , yp(x)=6x2ex , y(0)=6 , y(0)=0Find the unique solution satisfying the differential equation and the initial conditions given, where yp(x) is the particular solution. 81. y7y=4xe7x , yp(x)=27x2e7x449xe7x , y(0)=1 , y(0)=0Find the unique solution satisfying the differential equation and the initial conditions given, where yp(x) is the particular solution. 82. y+y=cosx4sinx , yp(x)=2xcosx+12xsinx , y(0)=8 , y(0)=4Find the unique solution satisfying the differential equation and the initial conditions given, where yp(x) is the particular solution. 83. y5y=e5x+8e5x , yp(x)=15xe5x+425e5x , y(0)=2 , y(0)=0In each of the following problems, two linearly independent solutions— y1 and y2 —are given that satisfy the corresponding homogeneous equation. Use the method of variation of parameters to find a particular solution to the given nonhomogeneous equation. Assume x0 in each exercise. 84. x2y+2xy2y=3x , y1(x)=x , y2(x)=x2In each of the following problems, two linearly independent solutions— y1 and y2 —are given that satisfy the corresponding homogeneous equation. Use the method of variation of parameters to find a particular solution to the given nonhomogeneous equation. Assume x0 in each exercise. 85. x2y2y=10x21 , y1(x)=x2 , y2(x)=x1Landing Vehicle NASA is planning a mission to Mars. To save money, engineers have decided to adapt one of the moon landing vehicles for the new mission. However, they are concerned about how the different gravitational forces will affect the suspension system that cushions the craft when it touches down. The acceleration resulting from gravity on the moon is 1.6 m/sec2, whereas on Mars it is 3.7 m/sec2. The suspension system on the craft can be modeled as a damped spring mass system. In this case, the spring is below the moon lander, so the spring is slightly compressed at equilibrium, as shown in Figure 7.12. Figure 7.12 The landing craft suspension can be represented as a damped spring-mass system. (credit “lander": NASA) We retain the convention that down is positive. Despite the new orientation, an examination of the forces affecting the lander shows that the same differential equation can be used to model the position of the landing craft relative to equilibrium: mx+bx+kx=0 , where m is the mass of the lander, b is the damping coef?cient, and k is the spring constant. 1. The lander has a mass of 15,000 kg and the spring is 2 m long when uncompressed. The lander is designed to compress the spring 0.5 m to reach the equilibrium position under lunar gravity. The dashpot imparts a damping force equal to 48,000 times the instantaneous velocity of the lander. Set up the differential equation that models the motion of the lander when the craft lands on the moon.Landing Vehicle NASA is planning a mission to Mars. To save money, engineers have decided to adapt one of the moon landing vehicles for the new mission. However, they are concerned about how the different gravitational forces will affect the suspension system that cushions the craft when it touches down. The acceleration resulting from gravity on the moon is 1.6 m/sec2, whereas on Mars it is 3.7 m/sec2. The suspension system on the craft can be modeled as a damped spring mass system. In this case, the spring is below the moon lander, so the spring is slightly compressed at equilibrium, as shown in Figure 7.12. Figure 7.12 The landing craft suspension can be represented as a damped spring-mass system. (credit “lander": NASA) We retain the convention that down is positive. Despite the new orientation, an examination of the forces affecting the lander shows that the same differential equation can be used to model the position of the landing craft relative to equilibrium: mx+bx+kx=0 , where m is the mass of the lander, b is the damping coef?cient, and k is the spring constant. 2. Let time t=0 denote the instant the lander touches down. The rate of descent of the lander can be controlled by the crew, so that it is descending at a rate of 2 m/sec when it touches down. Find the equation of motion of the lander on the moon.Landing Vehicle NASA is planning a mission to Mars. To save money, engineers have decided to adapt one of the moon landing vehicles for the new mission. However, they are concerned about how the different gravitational forces will affect the suspension system that cushions the craft when it touches down. The acceleration resulting from gravity on the moon is 1.6 m/sec2, whereas on Mars it is 3.7 m/sec2. The suspension system on the craft can be modeled as a damped spring mass system. In this case, the spring is below the moon lander, so the spring is slightly compressed at equilibrium, as shown in Figure 7.12. Figure 7.12 The landing craft suspension can be represented as a damped spring-mass system. (credit “lander": NASA) We retain the convention that down is positive. Despite the new orientation, an examination of the forces affecting the lander shows that the same differential equation can be used to model the position of the landing craft relative to equilibrium: mx+bx+kx=0 , where m is the mass of the lander, b is the damping coef?cient, and k is the spring constant. 3. If the lander is traveling too fast when it touches down, it could fully compress the spring and “bottom out.” Bottoming out could damage the landing craft and must be avoided at all costs. Graph the equation of motion found in part 2. If the spring is 0.5 m long when fully compressed, will the lander be in danger of bottoming out?Landing Vehicle NASA is planning a mission to Mars. To save money, engineers have decided to adapt one of the moon landing vehicles for the new mission. However, they are concerned about how the different gravitational forces will affect the suspension system that cushions the craft when it touches down. The acceleration resulting from gravity on the moon is 1.6 m/sec2, whereas on Mars it is 3.7 m/sec2. The suspension system on the craft can be modeled as a damped spring mass system. In this case, the spring is below the moon lander, so the spring is slightly compressed at equilibrium, as shown in Figure 7.12. Figure 7.12 The landing craft suspension can be represented as a damped spring-mass system. (credit “lander": NASA) We retain the convention that down is positive. Despite the new orientation, an examination of the forces affecting the lander shows that the same differential equation can be used to model the position of the landing craft relative to equilibrium: mx+bx+kx=0 , where m is the mass of the lander, b is the damping coef?cient, and k is the spring constant. 4. Assuming NASA engineers make no adjustments to the spring or the damper, how far does the lander compress the spring to reach the equilibrium position under Martian gravity?Landing Vehicle NASA is planning a mission to Mars. To save money, engineers have decided to adapt one of the moon landing vehicles for the new mission. However, they are concerned about how the different gravitational forces will affect the suspension system that cushions the craft when it touches down. The acceleration resulting from gravity on the moon is 1.6 m/sec2, whereas on Mars it is 3.7 m/sec2. The suspension system on the craft can be modeled as a damped spring mass system. In this case, the spring is below the moon lander, so the spring is slightly compressed at equilibrium, as shown in Figure 7.12. Figure 7.12 The landing craft suspension can be represented as a damped spring-mass system. (credit “lander": NASA) We retain the convention that down is positive. Despite the new orientation, an examination of the forces affecting the lander shows that the same differential equation can be used to model the position of the landing craft relative to equilibrium: mx+bx+kx=0 , where m is the mass of the lander, b is the damping coef?cient, and k is the spring constant. 5. If the lander crew uses the same procedures on Mars as an the moon, and keeps the rate of descent to 2 m/sec, will the lander bottom out when it lands on Mars?Landing Vehicle NASA is planning a mission to Mars. To save money, engineers have decided to adapt one of the moon landing vehicles for the new mission. However, they are concerned about how the different gravitational forces will affect the suspension system that cushions the craft when it touches down. The acceleration resulting from gravity on the moon is 1.6 m/sec2, whereas on Mars it is 3.7 m/sec2. The suspension system on the craft can be modeled as a damped spring mass system. In this case, the spring is below the moon lander, so the spring is slightly compressed at equilibrium, as shown in Figure 7.12. Figure 7.12 The landing craft suspension can be represented as a damped spring-mass system. (credit “lander": NASA) We retain the convention that down is positive. Despite the new orientation, an examination of the forces affecting the lander shows that the same differential equation can be used to model the position of the landing craft relative to equilibrium: mx+bx+kx=0 , where m is the mass of the lander, b is the damping coef?cient, and k is the spring constant. 6. What adjustments, if any, should the NASA engineers make to use the lander safely on Mars?Resonance Consider an undamped system exhibiting simple harmonic motion. In the real world, we never truly have an undamped system; -some damping always occurs. For theoretical purposes, however, we could imagine a spring-mass system contained in a vacuum chamber. With no air resistance, the mass would continue to move up and down indefinitely. The frequency of the resulting motion, given by f=1T=2 , is called the natural frequency of the system. If an external force acting on the system has a frequency close to the natural frequency of the system, a phenomenon called resonance results. The external forte reinforces and amplifies the natural motion of the system. 1. Consider the differential equation x+x=0 . Find the general solution. What is the natural frequency of the system?Resonance Consider an undamped system exhibiting simple harmonic motion. In the real world, we never truly have an undamped system; -some damping always occurs. For theoretical purposes, however, we could imagine a spring-mass system contained in a vacuum chamber. With no air resistance, the mass would continue to move up and down indefinitely. The frequency of the resulting motion, given by f=1T=2 , is called the natural frequency of the system. If an external force acting on the system has a frequency close to the natural frequency of the system, a phenomenon called resonance results. The external forte reinforces and amplifies the natural motion of the system. 2. Now suppose this system is subjected to an external force given by f(t)=5cost . Solve the initial-value problem x+x=5cost , x(0)=0 , x(0)=1 .Resonance Consider an undamped system exhibiting simple harmonic motion. In the real world, we never truly have an undamped system; -some damping always occurs. For theoretical purposes, however, we could imagine a spring-mass system contained in a vacuum chamber. With no air resistance, the mass would continue to move up and down indefinitely. The frequency of the resulting motion, given by f=1T=2 , is called the natural frequency of the system. If an external force acting on the system has a frequency close to the natural frequency of the system, a phenomenon called resonance results. The external forte reinforces and amplifies the natural motion of the system. 3. Graph the solution. What happens to the behavior of the system over time?Resonance Consider an undamped system exhibiting simple harmonic motion. In the real world, we never truly have an undamped system; -some damping always occurs. For theoretical purposes, however, we could imagine a spring-mass system contained in a vacuum chamber. With no air resistance, the mass would continue to move up and down indefinitely. The frequency of the resulting motion, given by f=1T=2 , is called the natural frequency of the system. If an external force acting on the system has a frequency close to the natural frequency of the system, a phenomenon called resonance results. The external forte reinforces and amplifies the natural motion of the system. 4. In the real world, there is always some damping. However, if the damping force is weak, and the external force is strong enough, real-world systems can still exhibit resonance. One of the most famous examples of resonance is the collapse of the Tacoma Narrows Bridge on November 7, 1940. The bridge had exhibited strange behavior ever since it was built. The roadway had a strange “bounce” to it. On the day it collapsed, a strong Windstorm caused the roadway to twist and ripple violently. The bridge was unable to withstand these forces and it ultimately collapsed. Experts believe the windstorm exerted forces on the bridge that were very close to its natural frequency, and the resulting resonance ultimately shook the bridge apart. This website (http://www.openstaxcollege.org/l/20_TacomaNarrow) contains more information about the collapse of the Tacoma Narrows Bridge. During the short time the Tacoma Narrows Bridge stood, it became quite a tourist attraction. Several people were on site the day the bridge collapsed, and one of them caught the collapse on film. Watch the video (http//www.openstaxcollege.org/l/20_TacomaNarr02) to see the collapse.Resonance Consider an undamped system exhibiting simple harmonic motion. In the real world, we never truly have an undamped system; -some damping always occurs. For theoretical purposes, however, we could imagine a spring-mass system contained in a vacuum chamber. With no air resistance, the mass would continue to move up and down indefinitely. The frequency of the resulting motion, given by f=1T=2 , is called the natural frequency of the system. If an external force acting on the system has a frequency close to the natural frequency of the system, a phenomenon called resonance results. The external forte reinforces and amplifies the natural motion of the system. 5. Another real-world example of resonance is a singer shattering a crystal wineglass when she sings just the right note. When someone taps a crystal wineglass or wets a finger and runs it around the rim, a tone can be heard. That note is created by the wineglass vibrating at its natural frequency. If a singer then sings that same note at a high enough volume, the glass shatters as a result of resonance. The TV show Mythbusters aired an episode on this phenomenon. Visit this website (http:// www.openstaxcollege.org/l/20_glass) to learn more about it. Adam Savage also described the experience. Watch this video (http://www.openstaxcollege.org/l/20_glass2) for his account.A mass weighing 4 lb stretches a spring 8 in. Find the equation of motion if the spring is released from the equilibrium position with a downward velocity of 12 ft/sec. What is the period and frequency of the motion?A mass weighing 2 lb stretches a spring 2 ft. Find the equation of motion if the spring is released from 2 in. below the equilibrium position with an upward velocity of 8 ft/sec. What is the period and frequency of the motion?A 100-g mass stretches a spring 0.1 m. Find the equation of motion of the mass if it is released from rest from a position 20 cm below the equilibrium position. What is the frequency of this motion?A 400-g mass swatches a spring 5 cm. Find the equation of motion of the mass if it is released from rest from a position 15 cm below the equilibrium position. What is the frequency of this motion?A block has a mass of 9 kg and is attached to a vertical spring with a spring constant of 0.25 N/m. The block is stretched 0.75 m below its equilibrium position and released. a. Find the position function x(t) of the block. b. Find the period and frequency of the vibration. c. Sketch a graph of x(t) . d. At what time does the block first pass through the equilibrium position?A block has a mass of 5 kg and is attached to a vertical spring with a spring constant of 20 N/m. The black is released from the equilibrium position with a downward velocity of 10 m/sec. a. Find the position function x(t) of the block. b. Find the period and frequency of the vibration. c. Sketch a graph of x(t) . d. At what time does the block first pass through the equilibrium position?A 1-kg mass is attached to a vertical spring with a spring constant of 21 N/m. The resistance in the spring- mass system is equal to 10 times the instantaneous velocity of the mass. a. Find the equation of motion if the mass is released from a position 2 m below its equilibrium position with a downward velocity of 2 m/sec. b. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.An 800-lb weight (25 slugs) is attached to a vertical spring with a spring constant of 226 lb/ft. The system is immersed in a medium that imparts a damping force equal to 10 times the instantaneous velocity of the mass. a. Find the equation of motion if it is released from a position 20 ft below its equilibrium position with a downward velocity of 41 ft/sec. b. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.A 9-kg mass is attached to a vertical spring with a spring constant of 16 N/m. The system is immersed in a medium that imparts a damping force equal to 24 times the instantaneous velocity of the mass. a. Find the equation of motion if it is released from its equilibrium position with an upward velocity of 4 m/sec. b. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.A l-kg mass stretches a spring 6.25 cm. The resistance in the spring-mass system is equal to eight times the instantaneous velocity of the mass. a. Find the equation of motion if the mass is released from a position 5 m below its equilibrium position with an upward velocity of 10 m/sec. b. Determine whether the motion is overdamped, critically damped, or underdamped.A 32-lb weight (1 slug) stretches a vertical spring 4 in. The resistance in the spring-mass system is equal to four times the instantaneous velocity of the mass. a. Find the equation of motion if it is released from its equilibrium position with a downward velocity of 12 ft/sec. b. Determine whether the motion is overdamped, critically damped, or underdamped.A 64-lb weight is attached to a vertical spring with a spring constant of 4.625 lb/ft. The resistance in the spring-mass system is equal to the instantaneous velocity. The weight is set in motion from a position 1 ft below its equilibrium position with an upward velocity of 2 ft/sec. Is the mass above or below the equation position at the end of sec? By what distance?A mass that weighs 8 lb stretches a spring 6 inches. The system is acted on by an external force of 8sin8t lb. If the mass is pulled down 3 inches and then released, determine the position of the mass at any time.A mass that weighs 6 lb stretches a spring 3 in. The system is acted on by all external force of 8sin(4t) lb. If the mass is pulled down 1 inch and then released, determine the position of the mass at any time.Find the charge on the capacitor in an RLC series circuit where L=40H , R=30 , C=1/200F , and E(t)=200V . Assume the initial charge on the capacitor is 7C and the initial current is 0A .Find the charge on the capacitor in an RLC series circuit where L=2H , R=24 , C=0.005F , and E(t)=12sin10tV . Assume the initial charge on the capacitor is 0.001C and the initial current is 0A .A series circuit consists of a device where L=1H , R=20 , C=0.002F , and E(t)=12V . If the initial charge and current are both zero, find the charge and current at time t.A series circuit consists of a device where L=12H , R=10 , C=150F , and E(t)=250V . If the initial charge on the capacitor is 0C and the initial current is 18A , find the charge and current at time t .Find a power series solution for the following differential equations. 104. y+6y=0Find a power series solution for the following differential equations. 105. 5y+y=0Find a power series solution for the following differential equations. 106. y+25y=0Find a power series solution for the following differential equations. 107. yy=0Find a power series solution for the following differential equations. 108. 2y+y=0Find a power series solution for the following differential equations. 109. y2xy=0Find a power series solution for the following differential equations. 110. (x7)y+2y=0Find a power series solution for the following differential equations. 111. yxyy=0Find a power series solution for the following differential equations. 112. (1+x2)y4xy+6y=0Find a power series solution for the following differential equations. 113. x2yxy3y=0Find a power series solution for the following differential equations. 114. y8y=0 , y(0)=2 , y(0)=10Find a power series solution for the following differential equations. 115. y2xy=0 , y(0)=1 , y(0)=3Find a power series solution for the following differential equations. 116. The differential equation x2y+xy+(x21)y=0 is a Bessel equation of order 1. Use a power series of the form y=n=0anxn to find the solution.True or False? Justify your answer with a proof or a counterexample. 117. If y and z are both solutions to y+2y+y=0 , then y+z is also a solution.True or False? Justify your answer with a proof or a counterexample. 118. The following system of algebraic equations has a unique solution: 6z1+3z2=84z1+2z2=4True or False? Justify your answer with a proof or a counterexample. 119. y=excos(3x)+exsin(2x) is a solution to the second-order differential equation y+2y+10=0 .True or False? Justify your answer with a proof or a counterexample. 120. To find the particular solution to a second-order differential equation, you need one initial condition.Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. 121. y2y=0Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. 122. y3y+2y=cos(t)Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. 123. ( dy dt)2+yy=1Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. 124. d2ydt2+tdydt+sin2(t)y=etFor the following problems, find the general solution. 125. y+9y=0For the following problems, find the general solution. 126. y+2y+y=0For the following problems, find the general solution. 127. y2y+10y=4xFor the following problems, find the general solution. 128. y=cos(x)+2y+yFor the following problems, find the general solution. 129. y+5y+y=x+e2xFor the following problems, find the general solution. 130. y=3y+xexFor the following problems, find the general solution. 131. yx2=3y94y+3xFor the following problems, find the general solution. 132. y=2cosx+yyFor the following problems, find the solution to the initial-value problem, if possible. 133. y+4y+6y=0 , y(0)=0 , y(0)=2For the following problems, find the solution to the initial-value problem, if possible. 134. y=3ycos(x) , y(0)=94 , y(0)=0For the following problems, find the solution to the boundary-value problem. 135. 4y=6y+2y , y(0)=0 , y(1)=1For the following problems, find the solution to the boundary-value problem. 136. y=3xyy , y(0)=3 , y(1)=0For the following problem, set up and solve the differential equation. 137. The motion of a swinging pendulum for small angles can be approximated by d2dt2+gL=0 , where is the angle the pendulum makes with respect to a vertical line, g is the acceleration resulting from gravity, and L is the length of the pendulum. Find the equation describing the angle of the pendulum at time t, assuming an initial displacement of 0 and an initial velocity of zero.The following problems consider the “beats” that occur when the forcing term of a differential equation causes “slow" and “fast” amplitudes. Consider the general differential equation ay+by=cos(t) that governs undamped motion. Assume that ba . 138. Find the general solution to this equation (Hint: call 0=b/a ).The following problems consider the “beats” that occur when the forcing term of a differential equation causes “slow" and “fast” amplitudes. Consider the general differential equation ay+by=cos(t) that governs undamped motion. Assume that ba . 139. Assuming the system starts from rest, show that the particular solution can be written as y=2a(022)sin(0t2)sin(0+t2) .The following problems consider the “beats” that occur when the forcing term of a differential equation causes “slow" and “fast” amplitudes. Consider the general differential equation ay+by=cos(t) that governs undamped motion. Assume that ba . 140. [T] Using your solutions derived earlier, plot the solution to the system 2y+9y=cos(2t) over the interval t=[50,50] . Find, analytically, the period of the fast and slow amplitudes.For the following problem, set up and solve the differential equations. 141. An opera singer is attempting to shatter a glass by singing a particular note. The vibrations of the glass can be modeled by y+ay=cos(bt) , where y+ay=0 represents the natural frequency of the glass and the singer is forcing the vibrations at cos(bt) . For what value b would the singer be able to break that glass? (Note: in order for the glass to break, the oscillations would need to get higher and higher.)