eigenvalue differential operator

Question

I stuck to this problem for half day and have no idea how to start. Can anyone please help me.

Find all the eigenvalues of the following equation

$(\frac{d}{dx}+x)(y(x))=\lambda$$y(x)$

I tried to let $y(x)=e^{\frac{-x^2}{2}}$

Then I get the eigenvalue of $0$, have no idea how to find the rest of the eigenvalues.

I also tried to multiply both sides by $e^{\frac{x^2}{2}}$, then I get

$\frac{d}{dx}[(e^{\frac{x^2}{2}})(y(x)]=\lambda$$y(x)e^{\frac{x^2}{2}}$

So is $0$ the ony eigenvalue?

Answer 1

Here's a hint. Try multiplying both sides by $e^{\frac{x^2}{2}} $. You can rework the operator on the left and you will get something much nicer.