What to use when reverse chain rule doesnt work?

[highc1157]

Hi there,

My equation to solve is (xy+(x^2))dx + (-1)dy=0

For method of exact solutions, the partials are not equal to each other so I cannot use
exact solutions (reverse chain rule)

I don't know how to solve this

 

[lurflurf]

What is reverse chain rule?
Any equation is exact in some coordinates so find them.
you stated (correctly)
(xy+x²)_y - (-1)_x
is not zero but we can use the fact that
(((xy+x²)_y -(-1)_x)/(-1))_y=0
to conclude that the equation is exact in
∫((xy+x²)_y - (-1)_x)/(-1)∂x
and y since (xy+(x²))dx + (-1)dy can be written
(-y-sqrt(-2(-x²/2)))d(-x²/2)-dy