$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 8.8: Digression to Differential Equations

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

Here is a standard use of series for solving differential equations.

## Example $$\PageIndex{1}$$

Find a power series solution to the equation

$f'(x) = f(x) + 2, \ \ \ \ \ f(0) = 0.$

Solution

We look for a solution of the form

$f(x) = \sum_{n = 0}^{\infty} a_n x^n.$

Using the initial condition we find $$f(0) = 0 = a_0$$. Substituting the series into the differential equation we get

$f'(x) = a_1 + 2a_2 x + 3a_3 x^3 +\ ... = f(x) + 2 = a_0 + 2 + a_1 x + a_2 x^2 + \ ...$

Equating coefficients and using $$a_0 = 0$$ we have

$\begin{array} {rclcr} {a_1} & = & {a_0 + 2} & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & {\Rightarrow a_1 = 2} \\ {2a_2} & = & {a_1} & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & {\Rightarrow a_2 = a_1/2 = 1} \\ {3a_3} & = & {a_2} & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & {\Rightarrow a_3 = 1/3} \\ {4a_4} & = & {a_3} & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & {\Rightarrow a_4 = 1/(3 \cdot 4)} \end{array}$

In general

$(n + 1) a_{n + 1} = a_n \ \ \ \Rightarrow \ \ \ a_{n + 1} = \dfrac{a_n}{(n + 1)} = \dfrac{1}{3 \cdot 4 \cdot 5 \cdot\cdot\cdot (n + 1)}.$

You can check using the ratio test that this function is entire.

8.8: Digression to Differential Equations is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.