exam2_ODE_solutions

exam2_ODE_solutions - Name UF ID number Exam 2 MAP 2302...

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Unformatted text preview: Name : UF ID number : Exam 2, MAP 2302, Fall’11 Show your work! Write your name on every piece of paper you turn in! 1 . Find a general solution of the equation y primeprime + 6 y prime + 13 y = 4 e- 3 x + 13 x 2 . Use the method of variation of parameters to find a particular solution of the Cauchy-Euler equation x 2 y primeprime- 3 xy prime + 4 y = ( x ln x ) 2 , x > 3 . Verify that y 1 = e x is a solution of the following differential equation and then use the reduction of order to find its general solution xy primeprime- (2 x + 1) y prime + ( x + 1) y = 0 , x > 4 . Let D = d/dx . Suppose that a differential equation has been simplified to the form D 3 ( D- 1) 2 ( D 2 + 4 D + 5) 2 y = 0 Find its general solution. 5 . Suppose that the displacement functions x ( t ) and y ( t ) for a coupled mass- spring system satisfy the initial value problem x primeprime ( t ) + 2 x ( t )- y ( t ) = 0 y primeprime ( t ) + 3 y ( t )- 2 x ( t ) = 0 , x (0) = x prime (0) = 0 , y (0) = 3 , y prime (0) = 0 Use the elimination method to solve for x ( t ) and y ( t ). 6 ( Extra credit ). Suppose that an external force f ( t ) is applied to the mass-spring system in Problem 5 so that the second equation is modified as y primeprime ( t ) + 3 y ( t )- 2 x ( t ) = f ( t ). If f ( t ) = g ( t ) sin(2 t ) where g ( t ) = 1 for 0 ≤ t ≤ 10 π and g ( t ) = 0 for t > 10 π . In other words, the harmonic force sin(2 t ) acts on the system only during the time 10 π , and then it is terminated. Describe and/or sketch a qualitative behavior of the displacement functions. Solutions of Exam 2, MAP 2302, Fall’11...
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exam2_ODE_solutions - Name UF ID number Exam 2 MAP 2302...

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