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.