Laplace transform of $f(t)=10te ^{-5t}$

Question

Find the Laplace transform of $$f(t)=10te ^{-5t}$$

Answer 1

We have:

$$\mathcal{L}(t) = \dfrac{1}{s^2}$$

Thus, we have:

$$\mathcal{L}(10 t e^{-5 t}) = 10 F(s+5) = \dfrac{10}{(s+5)^2}$$

Note: we made use of the Shifting Theorem, which was mentioned to you in another post.

Answer 2

HINT:

Use $$L\{e^{at}f(t)\}=F(s-a) \text{ if } L\{f(t)\}=F(s)$$

and $$L(t^n)=\frac{\Gamma(n+1)}{s^{n+1}}$$ and $$\Gamma(n+1)=n!\text{ if }n \text{ is non-negative integer}$$