I am confused on how to do partial fraction decomposition from a Laplace transform.
If you have $$\frac{d^2y}{dt^2} +3y = u_4(t)\cos(5(t-4)) \\ y(0)=0, y'(0)=-2$$
I can get to: $$L(y) = \dfrac{se^{4s}}{(s^2+25)(s^2+3)}-\dfrac{2}{(s^2+3)}$$
But I don't know how to do the partial fraction decomposition. Can someone explain?
$$f(s) = \frac{se^{-4s}}{(s^2+25)(s^2+3)}-\frac{2}{(s^2+3)}$$ $$f(s) =se^{-4s}( \frac{1}{(s^2+25)(s^2+3)})-\frac{2}{(s^2+3)}$$ You can decompose the fraction this way: $$f(s) =se^{-4s} \left( \frac{A}{(s^2+3)}+\frac{B}{(s^2+25)}\right)-\frac{2}{(s^2+3)}$$
You state : $$ A(s^2+25) + B(s^2+3) = 1 $$ You find that: $$A=-B=\frac 1 {22}$$ Therefore you have: $$f(s) =\frac 1 {22} \left( \frac{se^{-4s}}{(s^2+3)}-\frac{se^{-4s}}{(s^2+25)}\right)-\frac{2}{(s^2+3)}$$ Apply inverse Laplace Transform on each term.