Answer to Question #244746 in Differential Equations for Jac

Question

Solve the differential equation of the following exact differential equation. Show complete solution.

(3x²y - 6x)dx + (x³ + 2y)dy = 0

Answer 1

$\displaystyle \text{Let $M(x,y) = 3x^2y-6x$ }\\ \int (3x^2y-6x)dx = x^3y-3x^2+h(y)\\ \text{Differentiating and comparing to $N(x,y) = x^3+2y$, we have that}\\ h'(y) =2y \implies h(y) = y^2\\ \text{Hence the solution to the given exact equation is}\\ x^3y-3x^2+y^2=C$