Finding regular singular point

[BearY]

Homework Statement

Show that $x^2y''+sin(x)y'-y = 0$ has a regular singular point at $x=0$, determine the indicial equation and it's roots.

Homework Equations

For a DE in the form of $y''+p(x)y'+q(x)y=0$, if both $p(x)$ and $q(x)$ are not analytic at $x=x_0$, and both $(x-x_0)p(x)$ and $(x-x_0)^2q(x)$ are analytic at $x=x_0$,
$x_0$ is a singular regular point of the DE.

The Attempt at a Solution

I can't justify why $\frac{sin(x)}{x}$ is analytic at 0. Basically stucked at first step, so I am afraid no attempt worth mentioning.

Nevermind, I just realized that $\frac{sin(x)}{x}$ can be written as power series expension.

 

[Dick]

Homework Statement

Show that $x^2y''+sin(x)y'-y = 0$ has a regular singular point at $x=0$, determine the indicial equation and it's roots.

Homework Equations

For a DE in the form of $y''+p(x)y'+q(x)y=0$, if both $p(x)$ and $q(x)$ are not analytic at $x=x_0$, and both $(x-x_0)p(x)$ and $(x-x_0)^2q(x)$ are analytic at $x=x_0$,
$x_0$ is a singular regular point of the DE.

The Attempt at a Solution

I can't justify why $\frac{sin(x)}{x}$ is analytic at 0. Basically stucked at first step, so I am afraid no attempt worth mentioning.

Look at the Taylor series of $\sin(x)$. Divide it by $x$.