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Unformatted text preview: ODE, Review for Final Exam June 27, 2008 1. Find the inverse Laplace transform of X ( s ) = 4 s ( s 2 + 2 s + 2) . 2. What is the Laplace transform of the solution to y ′′ + 4 y ′ + 5 y = sin 2 t with initial condi tions y (0) = 1 and y ′ (0) = 2? 3. Compute the convolution product t * t . 4. Write down a function with Laplace transform s s 2 + 4 s + 5 . 5. For what values of c and k does the linear differential operator L = D 2 + cD + kI have unit impulse response given by w ( t ) = e − 2 t sin t (ie, L ( w ) = δ ( t ) ( I here means the identity linear differential operator, ie I ( y ) = y for all functions y ). 6. Consider a real 2 × 2 matrix A with eigenvalues 2 , 1 with corresponding eigenvectors parenleftbigg 1 2 parenrightbigg , parenleftbigg 2 1 parenrightbigg respectively. (a) Find the general solution to the system x ′ = A x and plot some solutions in the phase space....
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This note was uploaded on 10/18/2011 for the course MATH S3027D taught by Professor Peters during the Spring '11 term at Columbia.
 Spring '11
 Peters
 Differential Equations, Equations

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