quiz2_ODE - Name UF ID number Quiz 2 MAP 2302 Fall’11...

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Unformatted text preview: Name : UF ID number : Quiz 2, MAP 2302, Fall’11 Show your work! Write your name on every piece of paper you turn in! 1 . Find a particular solution of the equation y primeprime- y prime + 9 y = 3 sin(3 x ) 2 . Use the superposition principle to solve the initial value problem y primeprime- y = sin( x )- e 2 x , y (0) = 1 , y prime (0) =- 1 3 . Find a general solution of the Cauchy-Euler equation for x > 9 x 2 y primeprime + 15 xy prime + y = 0 4 . Find a second non-trivial solution of the following equation using reduc- tion of order if one solution is known: xy primeprime- ( x + 1) y prime + y = 0 , y 1 = e x , x > 5 . Use the method of variation of parameters to find a general solution of the following equation x 2 y primeprime + xy prime + 9 y = tan(3 ln( x )) Cheat: if y primeprime + py prime + qy = f , then the varied parameters are v 1 ( x ) =- integraldisplay fy 2 W dx, v 2 ( x ) = integraldisplay fy 1 W dx where W is the Wronskian of y 1 and y 2 that are two linearly independent solu- tion of the corresponding homogeneous equation. A further cheat: integraltext sec( u ) du = ln | sec( u ) + tan( u ) | 6 Extra Credit . The motion of an oscillator under the action of an external force is described by the equation y primeprime + ω 2 y = f ( t ) where y ( t ) is the position (amplitude) of the oscillator. Suppose that initially y (0) = y prime (0) = 0 and f ( t ) = f sin 2 ( ωt/ 2) with f = 10- 2011 . Will the amplitude y ( t ) exceed 2011,...
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quiz2_ODE - Name UF ID number Quiz 2 MAP 2302 Fall’11...

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