# differential Equations

laplace transform of f(t) = t^2 sin (at) use integration by parts

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1. L{t^2 sin(at)} = (-1)^2 F"(s)
where F(s) = a/(s^2+a^2)

To do the integration by parts, you will have to use multiple parts
u = t^2
dv = sin(at) e^-st
or
u = t^2 sin(at)
dv = e^-st dt

In the first case, you'll need to evaluate v using integration by parts, and the 2nd case will introduce more terms. But in either case, you need to get rid of the powers of t, then integrate by parts twice to get rid of the sin.

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