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\textbf{Math 365: Honors Analysis I}
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\textbf{Quiz 4}\quad \textsl{November 15, 2000}

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1. Let $L$ denote the Laplace transform. Prove:

a) $L[y']  = p L[y] - y(0)$

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b) $L[x f(x) ] = -F'(p)$ where $F(p) = L[f(x)]$.

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2. Solve the initial value problem  $y'' + 2y' + y = x^2$, $y(0) = 0$,
$y'(0) = 0$,

a) by direct application of Laplace transforms;

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b) by convolution using the impulsive response.

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