I have an interesting ODE and it involves multiplying polynomials. I like if anyone could kindly confirm that I am correct
$y'' + e^x y = 0$
$y(0) = 1$
$y'(0) = 2$
The goal was to find a solution in a series, but we only want five terms
What I did was basically wrote out the expansion for $e^x$ and my solution and set the related coefficients to 0. Note that my solution takes the form $y = \sum_{n=0}^{\infty} a_n x^n$
I can't neatly write out my work here because my align is different from \begin{align}. SO I unfortunately can only give you my "conclusion" to my final results. Sorry for this one, it's just impossible to write it all out neatly. But I hope you would believe that I did it cleanly on paper
I managed to reduced it to these set of equations
$a_0 + 2a_2 = 1$
$a_0 + a_1 + 6a_3 = 0$
$a_0/2 + a_1 + a_2 + 20a_4 = 0$
Resolving all the unknowns (with IC), I got $a_0 = 1, a_1 = 2, a_2 = \frac{-1}{2}, a_3 = \frac{-1}{2},a_4 = \frac{-1}{10}$
Also, I actually got an extra set of equation from matching the $x^4$ terms: $a_0/6 + a_1/2 + a_2 + a_3 +30a_5 = 0$, but that turned out redundant and I don't understand why.
So in conclusion I got $y \approx 1 + 2x -\frac{x^2}{2} -\frac{1}{2}x^3 - \frac{1}{10}x^4$
Thank you very much for reading