The differential operator is given by

$L= x\cdot \frac{d^2}{dx^2} + \frac{d}{dx} -\frac{a}{x}$

Any advice or strategies to find the eigenfunctions of this operator would be greatly appreciated. I initially thought it was cauchy-euler equation but

$Lu(x)=\lambda u(x)$


$xu''(x)+u'(x)-(\frac{a}{x}+\lambda) u(x)=0$

Changes the equation and I can no longer treat this as Euler-Cauchy and apply solution $u(x)=x^r$ to find the eigenfunctions. Could I perhaps put this in a more manageable form to find the eigenfunctions?


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