$\displaystyle x^2 \varphi'' (x)+x\varphi' (x)-\lambda\varphi(x)=0$

How is this DE solved?

This is a separation from PDE. I don't know what to do at this point.

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- Jan 3rd 2011, 07:32 PMdwsmithx^2 φ'' (x)+xφ' (x)-λφ(x)=0
$\displaystyle x^2 \varphi'' (x)+x\varphi' (x)-\lambda\varphi(x)=0$

How is this DE solved?

This is a separation from PDE. I don't know what to do at this point. - Jan 3rd 2011, 07:42 PMChris L T521
It's a Cauchy-Euler equation. You assume that the solution is of the form $\displaystyle \varphi(x)=x^r$. Thus, the characteristic equation for the ODE in question is $\displaystyle r^2-\lambda=0$.

I'm sure you can finish this off.