Perturbations method to solve differential equations

Question

Hi I am trying to solve the following differential equation $ \frac {d^2y }{dt^2}+ \epsilon \frac { dy}{dt}+1=0$ With the conditions $y (0)=0$ and $dy/dt(t=0)=1$ And what I got is
$y=- \frac {t^2}{2} + \epsilon \frac {t^4}{24} - \epsilon^2 \frac {t^6}{720}+ \dots$ But i am not sure if it is correct or not?

Answer 1

You can here compute the exact solution and compare your perturbation results against it. Integrating once gives $$ y'+ϵy+t=y_0'+ϵy_0+0=1 $$ This is still simple enough that the method of undetermined coefficients works, $$ y(t)=Ae^{-ϵt}+Bt+C\\ B+ϵ(Bt+C)=1-t\implies B=-\frac1ϵ,~C=\frac{1+ϵ}{ϵ^2}, \\ y_0=0\implies~ A=-C $$ so that finally $$ y(t)=\frac{(1+ϵ)(1-e^{-ϵt})-ϵt}{ϵ^2} =t-(1+ϵ)(\tfrac12t^2-\tfrac16ϵt^3+\tfrac1{24}ϵ^2t^4\mp...)\\ =t-\tfrac12t^2-ϵ(\tfrac12t^2-\tfrac16t^3)+ϵ^2(\tfrac16t^3-\tfrac1{24}t^4)\mp... $$