Question

The differential equation

Answer 1

As the question points out, the equation is indeed separable:

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{6x+9}{12y^2+16y+6}$

$\implies2(6y^2+8y+3)\,\mathrm dy=3(2x+3)\,\mathrm dx$

Integrate both sides to get

$2(2y^3+4y^2+3y)=3(x^2+3x)+C$

$\implies F(x,y)=\underbrace{(-3x^2-9x)}_{G(x)}+\underbrace{4y^3+8y^2+6y}_{H(y)}=C$