I have some problem with my equation: $$y''+5y = 5e^{-5t} $$
And we got that y(0) = 1 and y'(0) = 2
It's too much to write but what I get is: $$ \frac{e^{-5t}}{6}+1.516676089\cdot \cos(0.989 - \sqrt{5}t)$$ This works with first value y(0)=1 but doesn't work with y'(0).
Btw, where are supposed to use the complex roots and not the easy way with PFE, already solved this if used regular PFE.
Ignoring the Laplace transform, it's quite clear that the fundamental set of solutions to
$$y'' + 5y = 0$$
is
$$y_c = a_1 \sin \sqrt{5} t + a_2 \cos \sqrt{5} t$$
You can get the particular solution by just guessing
$$y_p = ae^{-5t}$$
Thus
$$y = a_1 \sin \sqrt{5} t + a_2 \cos \sqrt{5} t + a e^{-5t}$$
where you will find the coefficients by using the initial conditions, $y(0)=1$ and $y'(0) = 2$.