Find solution to exact ODE: $(4bxy(x)+3x+5)\dot{y}+3x^2+8ax+2by(x)^2+3y(x)=0\hspace{0,5cm}a,b\in\mathbb{R}$

Question

So I have this exact ODE

$(4bxy(x)+3x+5)\dot{y}+3x^2+8ax+2by(x)^2+3y(x)=0\hspace{0,5cm}a,b\in\mathbb{R}$

I already managed to show that $F=2bxy^2+3xy+5y+x^3+4ax^2$.

Now, how can I find the solution $y$ ? Seperating the variables is not really an option and it's also not linear. I'm kind of hitting a wall.

Answer 1

The solution to the ODE is $F=c\implies2bxy^2+(3x+5)y+(x^3+4ax^2)=c$, where $c$ is a constant. This is a quadratic equation in $y$ so you can use the quadratic formula to express $y$ as an explicit function of $x$, if it is really required.