---
title: "power series to solve differential equations"
type: concept
related: [Power Series, Simple Differential Equations, Diffeq Intro]
source: https://www.jemoka.com/posts/kbhpower_series/
confidence: high
status: active
---

We can now use power series to also solve differential equations.
\begin{equation} \dv{x}{t} = 0; x(0)=1 \end{equation}
We wish to have a power-series solution of shape:
\begin{equation} x(t) = \sum_{k=0}^{\infty }a_{k}t^{k} \end{equation}
We want to find the coefficients \(a_{k}\). If you can find such a function that fits this form, they both 1) converge and 20 behave the same way as \(e^{x}\) does in Simple Differential Equations.
analytic functions Functions which can be described with a power series are called analytic functions.
