Solve `((dy)/(dx))=e^(x-y)(e^x-e^y)dot`

Explanation

Multiplying the given equation by `e^(y)`, we get
`e^(y)(dy)/(dx) + e^(x)e^(y)=e^(2x)`
Putting `e^(y)=v`, so that `e^(y)(dy)/(dx) = (dv)/(dx)`, and equation (1) transform to `(dv)/(dx) +e^(x)v=e^(2x)`
I.F. `=e^(inte^(x)dx) = e^(e^(x))`
Hence, solution is `ve^(e^(x)) = int e^(2x)e^(e^(x))dx+c`
Let `e^(x)=t` or `e^(x)dx=intte^(t)dt+c`
or `e^(y)e^(e^(x))=e^(x)e^(e^(x))-e^(e^(x))+c`