Use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the convolution integral before transforming. (Write your answer as a function of s.) Lleft{t^2 cdot te^tright}

Question

Use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the convolution integral before transforming. (Write your answer as a function of s.)

L {t^{2} \cdot t e^{t}}

Answer 1

Step 1
Consider the provided question,
We have to find

L {t^{2} \cdot t e^{t}}

We have to use the convolution theorem:

L {f (t) \cdot g (t)} = L {f (t)} \cdot L {g (t)}

Step 2
Now, the given Laplace transform is find as,

L {t^{2} \cdot t e^{t}} = L {t^{2}} \cdot L {t e^{t}}

= \frac{2}{s^{3}} \cdot \frac{1}{(s - 1)^{2}}

(L {t^{n}} = \frac{n!}{(s^{n + 1})}, (L {t^{k} f (t) = (- 1)^{k} \frac{d^{k}}{d s^{k}} (L (f (t)))})

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

Use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the convolution integral before transforming. (Write your answer as a function of s.) Lleft{t^2 cdot te^tright}

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