Solving the Differential Equation ex y dy/dx = e-y + e-2x-y

[jofree87]

e^x y dy/dx = e^-y + e^-2x-y

Is this equation supposed to be solved through separation? I got an answer but it looks very messy. Can somebody check if I am doing this correctly?

e^x y dy/dx = e^-y + e^-2x-y

e^x y dy/dx = e^-y + e^-2x e^-y

e^x y dy/dx = e^-y ( 1 + e^-2x )

e^y y dy = e^-x ( 1 + e^-2x ) dx

Then Integrated by parts on both sides and got this solution:

e^y (y-1) = -e^-x - 1/3 e^-3x

Normally I would check it by plugging y and y' into the differential equation but I don't know how to take the derivative of y in this particular solution.

 

[tiny-tim]

Hi jofree87!

Normally I would check it by plugging y and y' into the differential equation but I don't know how to take the derivative of y in this particular solution.

Looks fine.

(and I'm afraid you can't always check by differentiating! )