I am trying to solve the PDE separation of variables problem for the PDE
$$\dfrac{-F''(x) + 2F'(x)}{F(x)} = \lambda$$
for the cases where $\lambda$ is negative, positive, and equal to 0.
I am stuck for the case where $\lambda = -\mu^2 < 0$:
$$-F''(x) + 2F'(x) - \lambda F(x) = 0$$
I was unable to solve this DE, because I couldn't solve the characteristic polynomial $-m^2 + 2M - \lambda = 0$. Please help!
If $\lambda=-\mu^2$ the characteristic equation is $$ -m^2+2\,m+\mu^2=0, $$ whose solution is $$ m=1\pm\sqrt{\mu^2+1}. $$