Solve the Differential liquation using separation of variabl

Solve the Differential liquation using separation of variables: e^x y dy/dx = e^-y + e^-2x - y

^{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)]

Divide by e^x

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

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

Divide by e^(-y)

[y/ e^(-y)] .dy/dx = e^(-x). e^(-3x)

y.e^(y).dy/dx = e^(-x). e^(-3x)

y. e^(y).dy = [ e^(-x). e^(-3x)] dx

Taking integration on both sides

INT [ e^y.y.dy = INT [ e^(-x) + e^(-3x)]dx

y.e^y - INT [ e^y].dy = -e^(-x) - (1/3).e^(-3x) + C

y.e^y - e^y = -e^(-x) - (1/3).e^(-3x) +C________ C is constant