# How tosolve this problem? (x-e^y)dy - dx = 0classify the equation: linear, nonlinear, separable,exact, homogeneous, or one that requires an integration factor.

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Expert Answers

sciencesolve | Certified Educator

You need to factor out `e^y` such that:

`e^y*(x/e^y - 1)dy - dx = 0`

You need to divide by `e^y` such that:

`(x/e^y - 1)dy - dx/(e^y) = 0`

Dividing by dx yields:

`(x/e^y - 1)dy/dx - 1/(e^y) = 0`

`` `(del f)/(del y)*(dy)/(dx) - (del f)/(del x) = 0`

You need to find a solution`f(x,y)=c` such that:`(del f)/(del x del y) = (del f)/(del y del x) = 1/e^y` .

`f(x,y) = int (del f)/(del y)*(dy) = int (x/e^y - 1)dy = x*e^(-y) - y + g(x)`

`` `(del f)/(del x) = e^(-y) + g'(x)` `f(x,y) = x*e^(-y) - y`

**Hence, the implicit solution to the differential equation is `x*e^(-y) - y = c =gt x = (c+y)*e^y.` **