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@jemoka / Jemoka Knowledge Base / raw/concept/kbhpower_series.md
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--- title: "power series to solve differential equations" source: https://www.jemoka.com/posts/kbhpower_series/ --- 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.