I am in a differential equation class and I am doing a project involving the perturbation method and this certain question is puzzling me. The question states.... Calculate the first order perturbation approximation to the initial value problem.
$y''+(1+ε \cos t)y=0$, $y(0)=1$, $y'(0)=0$
This is what I have done so far....
$y(t)=y_0+εy_1+ε^2 y_2+⋯$
$y''=y_0''+εy_1''+ε^{2} y_2''+⋯$
$y_0''+εy_1''+ε^2 y_2''+⋯(1+ε cost)(y_0+εy_1+ε^2 y_2+⋯)=0$
$y_0''+εy_1''+ε^2+y_2''+⋯+y_0+εy_1+ε^2 y_2+⋯+ε y_0cost+ε^2 y_1cost+ε^3 y_2 cost=0$
Then I ended up having all of the equations in which there is no $ε$, one $ε$, $ε^2$, and $ε^3$
$1→ y_0''+y_0=0$
$ε→ y_1''+y_1+y_0 cost=0$
$ε^2→ y_2''+y_2+y_1 cost=0$
$ε^3→ y_2 cost=0$
I know that I have to solve for each y, but I am unsure on how to go about it. I asked several other people and they don't know. Please help.
Thank you.