I can't remember if I ever did any of this stuff, but from a quick read of wikipedia and some other places (like

this), I think this is one way to do it:

1. Using a couple of facts (I assume that you don't need to prove everything from first principles):

Writing L(v) = F(s),

a) L(v') = sF(s) - v(0)

b) L(1) = 1/s

(if I've read correctly)

so, for v'+2v = 9:

L(v')+2(L) = L(9) (by linearity)

sF(s)+2F(s) = 9/s (we know that v(0) = 0)

F(s) = 9 / s(s+2)

Now you have to apply the inverse transform. I think that you're probably supposed to use tables here (not sure how to go about it otherwise).

This looks like it's going to be a trig function, but it needs to be put into the correct format, so (completing the square):

RHS = 9 / (s+1)^2 - 1

which (if I understand the tables correctly) is the transform of

**9 * e^(-t) * sinh(t)**
At this point, we can check back with the original equation:

Using various rules (product rule etc) and trig identities:

9 { -e^(-t) * sinh(t) + e^(-t) * cosh(t) } + {18 * e^(-t) * sinh(t)} = 9 (hopefully)

If you substitute the hyperbolics with their exponential formulae, I think this works (have to be careful with stray -1 factors).

Anyway, don't take my work for it. There could be mistakes in the above.