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Differential Equations

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Moderate

Series Solution about a Regular Singular Point

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Consider the equation $x^2y''+2xy'-(6+x)y=0$. $x=0$ is a regular singular point.

One can try solutions of the form $y=x^r\sum_{n=0}^{\infty}a_nx^n$, $a_0\neq 0$.

The equation that $r$ satisfies is called the indicial equation.

The roots of the indicial equation are the exponents.

Find the possible exponents at this regular singular point.

A

$-1+\sqrt{7}$ or $-1-\sqrt{7}$

B

$-\frac{1}{2}+\frac{1}{2}\sqrt{29}$ or $-\frac{1}{2}-\frac{1}{2}\sqrt{29}$

C

$-3$ or $2$

D

$3$ or $-2$