I feel super foolish asking this, but I've reached a mental block.
I'm trying to find the inverse laplace transform of:
$$\frac{s+3}{(s + 1)^2 (s-2)}$$
but, when I expand it into partial fractions:-
$$A(s+1)(s-2) + B(s-2) + C(s+1)^2$$
I get
$$B = -2/3$$ $$C = 5/9$$ $$A = -5/9$$
This is wrong, I've checked on wolfram alpha and it says the answer is
$$\\1/9{}{ e^{-t} (-6 t+5 e^{3t}-5)}$$
Please can you tell me where I'm going wrong?
The partial fraction expansion is:
$$\displaystyle \frac{s+3}{(s-2) (s+1)^2} = -\frac{5}{9(s+1)}-\frac{2}{3(s+1)^2}+ \frac{5}{9(s-2)}$$
The inverse Laplace Transform can now be done for each of those terms. We get:
$$\displaystyle \frac{1}{9} e^{-t} (-6 t+5 e^{3 t}-5)$$
Note that this is a somewhat easier way to display it. Of course, if you want to line things up with the partial fraction fractions, this is identically (just multiply each term in the above expression):
$$\displaystyle -\frac{5}{9} e^{-t} -\frac{2}{3} t e^{-t} + \frac{5}{9} e^{2 t}$$
Do you see it now? I think you may have done everything okay as your constants look correct.