So I have been given this tricky problem...
We are supposed to use the laplace transform formulas:
$L[f']=sF(s)-f(0)$
or
$L[f'']=s^{2}F(s)-sf(0)-f'(0)$
to find $F(s)$ when $f(t)=te^{kt}+\cosh(t)$
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Now, I THINK what we are supposed to do is to differentiate $f(t)$ so that you find
$f'(t)=kte^{kt}+\sinh(t);$
and
$f(0) = t+1;$
>
and then knowing what we do, sub it into $L[f']=sF(s)-f(0)$ so that we have:
$L[kte^{kt}+\sinh(t)]=sF(s)-(t+1)$
$\frac{k}{(s-k)^{2}}+\frac{1}{s^{2}-1}=sF(s)-(t+1)$
and then rearrange to find $F(s)$ so that:
$F(s) = \frac{k}{s(s-k)^{2}}+\frac{1}{s(s^{2}-1)}+\frac{1}{s}(t+1)$
what do you guys think??? yay or nay??? this seems a little too simple/easy and I feel like I am missing something... then again, I tend to over complicate things...