Question

Laplace Transform of Integral...

Answer 1

Solve
\(\sf f(t)=3t^2-e^{-t}-\int^{t}_{0}f(\tau)e^{t-\tau}d\tau\) for f(t)

I first have the laplace transform of each term and I ended up with:

\(\sf F(s)= \frac{6}{s^3}-\frac{1}{s+1}-F(s)•\frac{1}{s-1}\)

I am trying to solve this equation for F(s) and I got:

\(\sf F(s)= \frac{6}{s^3}-\frac{6}{s^4}-\frac{1}{s+1}+\frac{1}{s(s+1)}\)

which I think is wrong, since the book end up with something like:

\(\sf F(s)= \frac{6}{s^3}-\frac{6}{s^4}-\color{red}{\frac{1}{s}-\frac{2}{(s+1)}}\)

anyone can help me try understand how they end up with that? (I know I still have to get the inverse transform, but I'm pretty much stuck with that part for now).

Answer 2

Looks like they are using partial fractions decomposition to turn:

\[\frac{1}{s(s+1)} = \frac{1}{s} - \frac{1}{s+1}\]

I worked through the entire problem up to the same point as you and ended up with exactly the same answer as you so I'm pretty confident all your work to that point is correct.

Answer 3

oh okayyy, now i get it. Thank you so much!

Answer 4

Yeah glad I could help, keep up the good work!