Use the "mixed partials" check to see if the following differential equation is exact. If it is exact find a function F(x,y) whose level curves are solutions to the differential equation. (−3xy^2+y)dx + (−3x^2y+x)dy =0

We have[tex](-3xy^2+y)_y=--6xy+1[/tex]and[tex](-3x^2y+x)_x=-6xy+1[/tex]so the equation is indeed exact. So we want to find a function [tex]F(x,y)=C[/tex] such that[tex]F_x=-3xy^2+y[/tex][tex]F_y=-3x^2y+x[/tex]Integrating both sides of the first equation wrt [tex]x[/tex] gives[tex]F(x,y)=-\dfrac32x^2y^2+xy+f(y)[/tex]Differentiating both sides wrt [tex]y[/tex] gives[tex]F_y=-3x^2y+x=-3x^2y+x+f_y\implies f_y=0\implies f(y)=C[/tex]So we have[tex]F(x,y)=-\dfrac32x^2y^2+xy+C=C[/tex]or[tex]F(x,y)=\boxed{-\dfrac32x^2y^2+xy=C}[/tex]