Seperable Partial Differential Equation

[CINA]

Homework Statement

u_{x}=u_{y}+u

Homework Equations

Separation of variables

The Attempt at a Solution

It reduces to \frac{X'}{X}=\frac{Y'}{Y}+1

My question is does the 1 change at all how I use \lambda^{2}? In other words, will my solution still break down into this:

Case 1: \lambda^{2}=0

\frac{X'}{X}=0 and \frac{Y'}{Y}+1=0

Case 2: -\lambda^{2}

\frac{X'}{X}=-\lambda^{2} and \frac{Y'}{Y}+1=-\lambda^{2}

Case 3: \lambda^{2}

\frac{X'}{X}=\lambda^{2} and \frac{Y'}{Y}+1=\lambda^{2}

 

[jackmell]

This is correct although normally for first order PDEs, we use the method of characteristics.