Writing solution to an arbitrary ODE with arbitrary initial values as the sum of a power series?

Question

How can we solve for $y$ with these arbitrary initial values and polynomials? How would we write the solution as a power series?

Answer 1

Let $y(t)=\sum_{n=0}^\infty a_nt^n$, $f(t)=\sum_{n=0}^Ff_nt^n$ and $g(t)=\sum_{n=0}^Ga_nt^n$, where $F$ and $G$ are the degree of $f$ and $g$ respectively. Then \begin{align} y''(t)&=\sum_{n=0}^\infty(n+2)(n+1)a_{n+2}z^n\\ f(t)\,y'(t)&=\sum_{n=0}^\infty\Bigl(\sum_{i+j=n,\ j\le F}(i+1)a_{i+1}f_j\Bigr)z^n\\ g(t)\,y(t)&=\sum_{n=0}^\infty\Bigl(\sum_{i+j=n,\ j\le G}a_{i}g_j\Bigr)z^n \end{align} Plugging this into the equation we get if $n\ge D=\max(F,G)$ $$ (n+2)(n+1)a_{n+2}=-\sum_{i=n-F}^n(i+1)a_{i+1}f_{n-i}-\sum_{i=n-G}^na_{i}g_{n-i}. $$ The coefficient $a_{n+2}$ depends on the $D+2$ previous ones, an can be computed recursively, once $a_0$ and $a_1$ are fixed. To prove that $y(t)$ is in fact an entire function, you must get estimates on the coefficients $a_n$. I will not go into details. If $C=\max(|f_1|,\dots,|f_F|,|g_1|,\dots,|g_G|)$ it can be shown that $$ a_{n+2}\le\frac{C(D+2)}{n+2}\max_{n-D\le i\le n+1}|a_i|. $$ More estimates will prove that $\lim_{n\to\infty}|a_n|^{1/n}=0$.