Question

Find the Laplace transform of e^(-t)*t*sin(2t).

Answer 1

\[L \left\{ t^n f(t) \right\} = (-1)^n F^{(n)}(s)\]
where F^(n) is the nth derivative. and\[L \left\{ e^{at}f(t) \right\} = F(s - a)\]
we will combine these. in this case, n = 1 and a = -1

first find laplace of f(t) which is just sin(2t)
\[\frac{ 2 }{ s^2 + 4 }\]
cleaner to apply t^n rule before e^at rul, so we derive once.
\[-\frac{ 4s }{ (s^2 + 4)^2 }\]apply e^at rule, all s become s+1\[-\frac{ 4(s+1) }{ ((s+1)^2 + 4)^2 }\]usually no need to simplify

Answer 2

made small mistake. final answer is positive because of the (-1)^n which i forgot about

Answer 3

Thanks so much!

Answer 4

glad i could help :P i find these so fun

Answer 5

Lol I'm not there yet, but it's growing on me :)