What is the method for solving the eigenvalue problem with integration by parts?

[MagicQuantum]

Solve the eigenvalue problem O$$_{6}$$ $$\Psi$$(x) = $$\lambda$$ $$\Psi$$(x)

O$$_{6}$$$$\Psi$$(x) = $$\int$$ from negative infinity to x of dxprime *$$\Psi$$(xprime) * xprime
what values of eigenvalue $$\lambda$$ lead to square integral eigenfuctions? (Hint: Differentiate both sides of the equation with respect to x)

Im trying to do this with integration by parts but i keep getting infinity in some form or another. I am assuming $$\Psi$$(xprime) is equal to the derivative of $$\Psi$$(x) with respect to x. so i end up with $$\lambda$$$$\Psi$$(xprime) is equal to some integral that keeps working out to infinity. I don't want anyone to give me a solution but if anyone can give me a bump in the right direction i would be pumped.

Thanks

 

[Dick]

You aren't paying enough attention to the hint. How do you find d/dx of an integral with x as the upper limit? You should get a simple ODE to solve for psi.