# Differential equation given integrating factor

1. ### naspek

181
1. The problem statement, all variables and given/known data

Show that given function μ is an integrating factor and solve the differential equation..

y^2 dx + (1 + xy) dy = 0 ; μ(x) = e^xy

3. The attempt at a solution

let M = y^2
N = (1 + xy)

dM/dy = 2y dN/dx = y hence, not exact equation.

times μ(x) = e^xy to the not exact equations...

2y(e^xy) dx + y(e^xy) dy = 0

let M = 2y(e^xy)
N = y(e^xy)

dM/dy = 2(e^xy) + 2y(e^y) ---> apply product rule

dN/dx = 0(e^xy) + y(e^y) ---> apply product rule

the problem is.. the equations still not the exact equations..
How to proceed?

2. ### HallsofIvy

41,265
Staff Emeritus
Yes- which means that $\mu= e^{xy}$ is NOT an integrating factor. Something is wrong with that question.

3. ### naspek

181
So... i can't solve this equation? the equation doesn't have any solutions?