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**Homework Statement**

**The attempt at a solution**

I'm using the method of separation of variables by first defining the solution as [itex]u(x,t) =X(x)T(t)[/itex]

Putting this back into the PDE I get: [itex]T''X = x^{2}X''T + xX'T[/itex]

which is simplified to [tex]\frac{T''}{T} = \frac{x^{2}X'' + xX'}{X} = -\lambda^{2}[/tex]

The spatial problem is then: [itex]x^{2}X'' + xX' = -X\lambda^{2}[/itex]

Is this correct so far? How do I continue?