La place Transform

SteelyDanJalapeno

I'm trying to find the L transform of tsinh(4t)

I've couple of techniques and i've got the transform of each function seperatly, for "t" and "sinh(4t)" but i've recently looked at the convolution theorem which looks at the multiplication of these,

so my question is, i was asked to get the laplace transform of tsinh(4t) is the convolution theorem the only way to do this?

or should i leave the answer as two laplace transforms one of "t" and one of sinh(4t), which is so much easier.

i'm not even sure if its possible to get a convolution of these, feeling quite confused here

Clinker

You want the differentiation theorem:

If [latex]L\{f(t)\} = F(s)[/latex] then [latex]$\displaystyle L\{tf(t)\} = - \frac{d}{ds}F(s)$[/latex]

under suitable conditions. So find the LT of sinh(4t) and differentiate it w.r.t. s, then multiply by - 1.