Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2013 / Differential Equations / 9 / b / Solution

Integrating the complex exponential and taking its even and odd combinations gives the standard Laplace transforms

L [a_{1}] (p) = \frac{p}{p ^{2} + ω ^{2}}, L [a_{2}] (p) = \frac{ω}{p ^{2} + ω ^{2}} .

For example,

L [e^{iω t}] = (p - iω)^{- 1}

for

Re p > 0

, and the real sine and cosine identities follow by combining the two signs of

iω

.

Apply part (a) once to the cosine transform and twice to the sine transform:

L [a_{3}] (p) L [a_{4}] (p) = - \frac{d}{d p} \frac{p}{p ^{2} + ω ^{2}} = \frac{p ^{2} - ω ^{2}}{( p ^{2} + ω ^{2} ) ^{2}}, = \frac{d ^{2}}{d p ^{2}} \frac{ω}{p ^{2} + ω ^{2}} = \frac{2 ω ( 3 p ^{2} - ω ^{2} )}{( p ^{2} + ω ^{2} ) ^{3}} .

All four transforms are valid on the stated right half-plane.

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