Given a differential equation $$xy''+y'+xy=0~.$$ I have to find the number of linearly independent solutions about $x=0$.

My attempt is to find the Wronskian at $x=0$ which is infinity. I cannot understand how to proceed.

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    $\begingroup$ This is a well-studied DifEq and has solution $y(x) = c_1 J_0(x) + c_2 Y_0(x)$ in terms of Bessel Functions. You can show this, e.g., by supposing $y$ is analytic and working with the power series. Knowing this, you should get two linearly independent solutions. $\endgroup$ – Brevan Ellefsen Sep 9 '18 at 10:40
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    $\begingroup$ Another thing to try (if you do not know about Bessel functions) is series solution of the DE. $\endgroup$ – GEdgar Sep 9 '18 at 10:54

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