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How do I solve the following differential equation?

$\partial^{2}_{x} \left[x^{2}p\right] + \partial_{x} \left[\left(x-1\right)p\right] = 0$

I tried a Fourier transform which leads to

$\left[k^{2}\partial^{2}_{k} + k\left(\partial_{k}+i\right)\right]\tilde{p} = 0$

where $\tilde{p}$ is the Fourier transform of $p$ but that doesn't really help.

Any ideas?

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    It's like a Cauchy Euler problem, modulo that pesky $i$ term. I think you might be able to solve via series method on the Fourier transform.2012-10-12
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    Expand it first.2012-10-12
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    Hm, if I expand $p$ into a power series, I get a problem with the radius of convergence.2012-10-12
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    Would this question be suitable for mathoverflow?2012-10-15
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    Okay, technically the above differential equation has an irregular singularity at x=0. That is why e.g. the Frobenius method does not work. Any ideas how to proceed from here?2012-10-15
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    The "expand" I mean is that simplify the DE to the standard form rather than assuming power series solution. Does the operator $\partial_x^2$ in this question means $\dfrac{\partial^2}{\partial x^2}$ or $\left(\dfrac{\partial}{\partial x}\right)^2$ ?2012-10-15
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    It means the second derivative. If I expand the above equation, I get $x^{2}p^{\prime\prime} + (5x-1)p^{\prime} + 3p = 0$ which has an irregular singular point at $x=0$.2012-10-15

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