So I just started in differential equation. I ran into some trouble with a problem in an exercise.
The problems goes:
Find the first four nonzero terms in each of two power series solutions about the origin.
$e^xy'' + xy = 0$
since $e^x$ is analytic at every real number, thus I know the power series solution exist at $x=0$
But I do not know how to find them. Hence I was hoping using this question as an example, perhaps someone could give me a general steps/guide to solve similar questions?
P.S: This exercise is from the book Elementary differential equation and Boundary problem.
Thank you in advance for any help given
Best Answer
Let $y(x)=\sum_{k\geq 0}a_kx^k$, then $$y''(x)=\sum_{k\geq 2}k(k-1)a_kx^{k-2}=\sum_{k\geq 0}(k+2)(k+1)a_{k+2}x^{k}=2a_2+6a_3 x+12a_4 x^2+o(x^2).$$ Moreover $$e^x=\sum_{k\geq 0}\frac{x^k}{k!}=1+x+\frac{x^2}{2}+\frac{x^3}{6}+o(x^3).$$
Plug all these series in $e^xy''(x) + xy(x)$. The coefficients of resulting series should be all zero.
We have that the first few terms of the expansion are: $$e^xy''(x) + xy(x)=2a_2+(6a_3+2a_2+a_0)x+(12a_4+6a_3+a_2+a_1)x^2+o(x^2)$$ which implies that $$\mbox{$a_2=0$, $6a_3+2a_2+a_0=0$, $12a_4+6a_3+a_2+a_1=0$.}$$ The coefficients $a_n$ are not uniquely determined because the differential equation admits infinite solutions.