Eigenvalue of first order nonlinear differential equation

Question

How to solve for the eigenvalues of equation $$(\partial_x+1-iay)y=\lambda y$$ where y is a function of x, a is a real parameter, $\lambda$ is the eigenvalue.

Answer 1

$y' + (1-iay)y = \lambda y$

$y' = \left((\lambda-1 )+iay\right)y$

$\frac{1}{\left((\lambda-1 )+iay\right)y}\frac{\partial y}{\partial x} = 1$

$\left[\frac{1}{y}+\frac{-ia}{(\lambda-1 )+iay}\right]\frac{\partial y}{\partial x}=\lambda-1$

$\ln y - \ln\left((\lambda - 1)+iay)\right)=(\lambda-1)x + C$

$\frac{y}{(\lambda - 1)+iay}=Ce^{(\lambda-1)x}$

$y=\frac{(\lambda - 1)Ce^{(\lambda-1)x}}{1-iaCe^{(\lambda-1)x}}$