Solve DE with Substitution: y' = cos(x-y)

[manenbu]

Homework Statement

Solve:
y' = cos(x-y)

Homework Equations

The Attempt at a Solution

Using x-y=t and solving the integrals, I get that the general solution is:
-cot(0.5(x-y)) = x + c which is correct, but there's another solution which is x-y=2πk, but I don't understand why.
In the integral I get that it is undefined for 1-cost=0.
Meaning it is undefined for cost = 1, or cos(x-y) = 1.
Plugging it in the original problem to see if it satisfied it gives me y' = 1. How can I know if this is true for all y (I guess it is)...
I didn't exactly understand what do I need to do here.

 

[Pere Callahan]

if you want to check that y'=1 you best start with the definition of y...which is just x-2πk...