Question

find a general solution in powers of x of the differential equation. state the recurrence relation and the guaranteed radius of convergence.

$(x^{2}-25)y + 4xy + 2y = 0$

the recurrence relation is $c_{n + 2}=\frac{c_{n}}{25}$.

find a general solution in powers of x.

$y(x)=\square$

(type an expression in terms of $c_{0}$ and $c_{1}$. type any series in summation notation using n as the index variable and 0 as the starting index.)

Explanation:

Step1: Analyze the recurrence relation

Given \(c_{n + 2}=\frac{c_{n}}{25}\).
For even - numbered coefficients (\(n = 2k,k = 0,1,2,\cdots\)):
When \(n = 0\), \(c_{2}=\frac{c_{0}}{25}\); when \(n = 2\), \(c_{4}=\frac{c_{2}}{25}=\frac{c_{0}}{25^{2}}\); in general, \(c_{2k}=\frac{c_{0}}{25^{k}}\).
For odd - numbered coefficients (\(n=2k + 1,k = 0,1,2,\cdots\)):
When \(n = 1\), \(c_{3}=\frac{c_{1}}{25}\); when \(n = 3\), \(c_{5}=\frac{c_{3}}{25}=\frac{c_{1}}{25^{2}}\); in general, \(c_{2k + 1}=\frac{c_{1}}{25^{k}}\).

Step2: Write the power - series solution

The power - series solution \(y(x)=\sum_{n = 0}^{\infty}c_{n}x^{n}=c_{0}+c_{1}x + c_{2}x^{2}+c_{3}x^{3}+\cdots\)
Substitute the expressions for \(c_{n}\):

$$
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$$

Using the formula for the sum of a geometric series \(\sum_{k = 0}^{\infty}r^{k}=\frac{1}{1 - r}\) (for \(|r|\lt1\)), where \(r=\frac{x^{2}}{25}\)

Answer:

\(y(x)=c_{0}\sum_{k = 0}^{\infty}(\frac{x^{2}}{25})^{k}+c_{1}x\sum_{k = 0}^{\infty}(\frac{x^{2}}{25})^{k}\)