Quiz1 - to find the general solution of the nonhomogeneous...

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April 12, 2007 MAE 105 Quiz #1 Total Time: 20 minutes PROBLEM 1 a. (0.5 Point) Solve the following ODE by twice integration, and find its general solution (which must include two integration constants): dx 2 d 2 u ( x ) = sin 2 x , 0 < x < π . b. (0.5 Point) Cosider the following boundary conditions: u (0) = 2 , dx du ( π ) = - 1 , for the above ODE, and find the final solution that satisfies both the ODE and the boundary conditions. PROBLEM 2 Consider the following ODE y ′ - 3 y = f ( x ) , x > 0 , (*) and the initial condition y (0) = 3 , (**) where prime denotes differentiation with respect x and f is a given function of x . 1. (0.5 Point): Write down the corresponding homogeneous equation. 2. (1 Points): Find the general solution of the homogeneous equation. [Note: Your general solution must include a constant of integration.] 3. (1 Point): Use the method of variation of parameter
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Unformatted text preview: to find the general solution of the nonhomogeneous equation (*) in an integral form. [Note: Your general solution must include a new constant of integration.] 4. (1 Point): For f ( x ) = x , use integration-by-parts to find the final solution explicitly. 5. (0.5 Point): Use the initial condition (**) to obtain the integration constant and write down the explicit final solu-tion. 6. (Extra Point): For an extra point, find the complete solution if f ( x ) = sin x and the same initial condition. [Note: You must give the expression for y as an explicit function of x .] Note: To receive full credit, all steps must be neatly shown, following the requested procedure. Writing down the final results will receive no credit....
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