8.8: Digression to Differential Equations

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

Example \(\PageIndex{1}\)

Find a power series solution to the equation

\[f'(x) = f(x) + 2, \ \ \ \ \ f(0) = 0. \nonumber \]

Solution

We look for a solution of the form

\[f(x) = \sum_{n = 0}^{\infty} a_n x^n. \nonumber \]

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

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

Equating coefficients and using \(a_0 = 0\) we have

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

In general

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

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