I'm trying to understand how to predict function behavior of the solution curves for differential equations. In this case I'm following along with an example in my textbook of a separable differential equation:
$$dy/dx=\frac{4-2x}{3y^2-5}$$
From that we know the solution curve has asymptotes at +${\sqrt \frac{5}{3}}$ & -${\sqrt \frac{5}{3}}$. I've determined the implicit solution to be... $$y^{3}-5y=\left(4x-x^{2}\right)+C$$
...and after rearranging it, I know C describes the difference between the functions. $$y^{3}-5y-\left(4x-x^{2}\right)=C$$
What I don't understand is how I could possibly predict the solution curve changes shape when C<0.4. What about the fact that the top portion disappears when C<8.3? Is there someway to do this other than plug & chug values?
Here's the solution graph given by the book :
