General solution of a differential equation

[Briggs]

Homework Statement

Find the general solution to the differential equation in implicit form.
http://www.texify.com/img/%5Clarge%5C%21%5Cfrac%7Bdy%7D%7Bdx%7D%3D%28cosx-sinx%29e%5E%7Bcosx%2Bsinx%2By%7D.gif

Homework Equations

http://www.texify.com/img/%5Clarge%5C%21%5Cfrac%7Bdy%7D%7Bdx%7D%3Df%28x%29g%28y%29.gif

The Attempt at a Solution

Am I correct in assuming that this is a separation of variables problem? I can't seem to grasp the algebra to move the y to the left hand side. Could anybody give me a nudge in the right direction?

 

[rock.freak667]

split exp{cosx+sinx+y} into exp(cosx+sinx)*exp(y)

 

[Architeuthis Dux]

Yes, this is a separation of variables problem. To solve it, you need to separate the variables x and y onto opposite sides of the equation. This can be done by multiplying both sides by dx and dividing both sides by (cosx-sinx)e^(cosx+sinx+y). This will give you the following equation:

dy/(cosx-sinx)e^(cosx+sinx+y) = dx

Now, you can integrate both sides with respect to their respective variables. The left side can be integrated using the substitution u = cosx+sinx+y, which will give you du = (-sinx+cosx+1)dx. The right side can be integrated using the substitution v = cosx-sinx, which will give you dv = (-sinx-cosx)dx.

After integrating and substituting back in the original variables, you will get:

ln|cosx+sinx+y| = ln|cosx-sinx| + C

Where C is the constant of integration. This can be rewritten as:

cosx+sinx+y = C(cosx-sinx)

Which is the general solution to the given differential equation in implicit form.