**1. Homework Statement**

Identify the type of singularity at x=0 for these differential equations

x*Sin[1/x]*y''[x]+y[x]==0

x^2*y''[x]+Sin[1/x]*y[x]==0

**2. Homework Equations**

A Singular point is regular if f(x)(x-x_0)^n is defined as x approaches x_0 and is analytic in a near a neighborhood of that singular point. It is irregular if this doesn't hold.

**3. The Attempt at a Solution**

x/(Sin[1/x]) x is analytic at x=0 and around x=0 to see if this was a regular or irregular singularity. So I wrote out the Taylor series of Sin[x] and plugged in for x 1/x. After doing the it appeared that as x goes to zero x/Sin[1/x] went to zero so I said it was a regular singularity, I also identified other singular points at 1/(Pi*n) where n is a integer. Using the same process for the other ODE I found that as x goes to zero (x^2/(Sin[1/x)]))x^2 does not converge as x goes to zero thus it is a irregular singularity. Can anyone tell me if my approach was right and if my answers were right?

x/(Sin[1/x]) x is analytic at x=0 and around x=0 to see if this was a regular or irregular singularity. So I wrote out the Taylor series of Sin[x] and plugged in for x 1/x. After doing the it appeared that as x goes to zero x/Sin[1/x] went to zero so I said it was a regular singularity, I also identified other singular points at 1/(Pi*n) where n is a integer. Using the same process for the other ODE I found that as x goes to zero (x^2/(Sin[1/x)]))x^2 does not converge as x goes to zero thus it is a irregular singularity. Can anyone tell me if my approach was right and if my answers were right?