Diff. Eq. With Power Series

Question

Is there a way to solve differential equations using power series?

Also, can you represent a derivative as a sum?

Answer 1

Assume you are given an IVP $$y'(t)=f\bigl(t,y(t)\bigr),\quad y(0)=0\ ,\tag{0}$$ where the given right hand side $f(t,y)$ can be expanded into a double power series in its variables: $$f(t)=\sum_{j\geq 0, \>k\geq 0} c_{jk}t^j y^k\qquad\bigl(|t|<\delta, \>|y|<\delta\bigr)\ .\tag{1}$$ Then you can make the "Ansatz" $$y(t)=\sum_{r\geq1} a_r t^r\tag{2}$$ with unknown coefficients $a_r$ $(r\geq1)$. Plugging this "Ansatz" into $(1)$ and comparing coefficients of $t^r$ for $r=1, \ 2,\ 3,\ldots$ you will obtain finitary recursion formulas for the $a_r$ in terms of the $a_l$ with $l<r$. The resulting power series $(2)$ then is indeed the solution to the given IVP $(0)$.