Solve the differential equation?

Question

(3 x^2 y^2 - y^3 + 2 x) dx + (2 x^3 y - 3 x y^2 + 1) dy =0

Answer 1

(3 x² y² - y³ + 2 x) dx + (2 x³ y - 3 x y² + 1) dy =0

The given differential equation is in the form of M dx + N dy =0 .

M = 3 x² y² - y³ + 2 x and N = 2 x³ y - 3 x y² + 1

Check for exact differential equation .

Necessary and sufficient condition for the differential equation Mdx + Ndy = 0 to be exact is   .

So .

So the given differential equation is exact differential equation .

So the general solution for exact is  .

Where g'(y) is the terms not containing x in N .

So g '(y) = 1

So the general solution is   x³y² + x² +y +c  .