How to solve it Verify that the following differential equa

How to solve it ?

Verify that the following differential equation is exact, then solve. (1 + ye^xy)dx + (2y + xe^xy)dy = 0

Solution

Given Equation -

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

To Prove- Given Equation is Exact

Proof-

It is of format M(x,y) dx + N(x,y) dy=0

For a Equation to be \"Exact\"

  dM/dy =dN/dx

So, Calculating the values:

dM/dy = e^(xy) + (yx)e^(xy)----------------------------------------------[Equation(1)]

dN/dx = e^(xy) + (yx)e^(xy)-----------------------------------------------[Equation(2)]

Hence, Equation(1) =Equation(2)

The Given Differential Equation is Exact

PROVED