i've been working on this question for a while and can't seem to find the solution
Use Laplace transforms to solve the ODE $$y''(t) + 2y'(t) + 5y(t) = e^{-t}$$
the initial conditions are y(0) = 1 and y'(0) = 1.
so far this is how I have progressed:
so generally $$y''+ay'+by=\mu$$ gives
$$(S^2Y-sy(0)-y'(0))+a(SY-y(0))+bY=\mu$$
this equals
$$(s^2 + as + b)(Y(s))=(s+a)y(0) + y'(0) + \mu$$
so using my equation i plugged in my values getting
$$(s^2 + 2s + 5)(Y(s))=(s+2)(1)+ 1 + (1/(s+1))$$
now i understand that you need to bring thee quadratic over and simplify so that you can do partial fractions. however my quadratic doesn't factorise, so i don't understand how else i can continue.
any help would be veryu appreciated. Thank you