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.