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The two equations at hand are $$x\sin\left(\frac 1x\right)y''(x)+y(x)=0 \\ x^2y''(x)+\sin\left(\frac 1x\right)y(x)=0$$ For the first equation I wrote $y''(x)+\dfrac{y(x)}{\sin\left(\frac 1x\right)x}$ and then went and tested to see if $\dfrac x{\sin\left(\frac 1x\right)x}$ 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 $x \frac 1x$.

After doing that it appeared that as $x$ goes to zero $\dfrac x{\sin\left(\frac 1x\right)}$ went to zero so I said it was a regular singularity, I also identified other singular points at $\dfrac 1{\pi n}$ where $n$ is a integer. Using the same process for the other ODE I found that as $x$ goes to zero $x^2\dfrac{\sin\left(\frac 1x\right)}{x^2}$ 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?

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  • $\begingroup$ Please use LaTeX to format your maths text. See here. $\endgroup$ – Mattos Nov 13 '16 at 2:02

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