x^2 φ'' (x)+xφ' (x)-λφ(x)=0

**dwsmith** · January 3rd 2011, 07:32 PM

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

How is this DE solved?

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

**Chris L T521** · January 3rd 2011, 07:42 PM

Originally Posted by dwsmith

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

How is this DE solved?

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

It's a Cauchy-Euler equation. You assume that the solution is of the form $\varphi(x)=x^r$ . Thus, the characteristic equation for the ODE in question is $r^2-\lambda=0$ .

I'm sure you can finish this off.

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x^2 φ'' (x)+xφ' (x)-λφ(x)=0