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Unformatted text preview: REVIEW PROBLEMS FOR THE FIRST MIDTERM 1. (a) Find the inverse transform of ( s 2) e s s 2 4 s + 3 . (b) Compute the Laplace transform of δ ( t 5) + g ( t ), where g ( t ) = sin t for ≤ t < 2, g ( t ) = 0, for t ≥ 2. (Note: it may be helpful to use sin( θ + ψ ) = sin( θ )cos( ψ ) + sin( ψ )cos( θ ). (c) Find the Laplace transform of R t f ( τ ) dτ , where L{ f } ( s ) = 1 / ( s 2 + 1) 3 / 2 . 2. (a) Find the Laplace transform of y if y 00 + 2 y + y = f ( t ), y (0) = 1, y (0) = 0, where f ( t ) = 1 if 0 ≤ t < 1 and f ( t ) = 0, if t ≥ 1. (b) Without inverting the transform explicitly, which of the following terms might appear in the solution y ( t )? Which terms could not possibly appear in the solution y ( t )? Justify your answer. (Hint: To invert the Laplace transform of y , you might need to set up some partial fraction decompositions. But to answer this question you do not need to compute the numerical values of the coefficients in these decompos tions.) (i) ce...
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 Fall '06
 DanOcone
 Laplace, Complex differential equation, Frobenius method, Regular singular point, indicial equation

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