Question

Determine the Laplace transform of

f(t)=t^2e^2t

Answer 1

Answer:

We are given: $f(t) = t^2e^{2t}$

We need to find out the Laplace transform of $f(t)=t^2e^{2t}$

Apply Laplace transformation rule, where $f(t)=e^{2t},k=2$

We need to compute $\frac{d^2}{ds^2}(\frac{1}{s-1})$

Solve by using the chain rule

$\frac{d}{ds}(\frac{1}{s-1})=-\frac{1}{(s-2)^2}\frac{d}{ds}(s-2)=-\frac{1}{(s-2)^2}$

Solve by using the chain rule

$\frac{d^2}{ds^2}(\frac{1}{s-2})=\frac{d^2}{ds^2}(-\frac{1}{(s-2)^2})=\frac{2}{(s-2)^2}\frac{d}{ds}(s-2)=\frac{2}{(s-2)^3}$

Hence $\fbox{L\{$t^2e^{2t}$\}=$\frac{2}{(s-2)^3}$}$

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