Quadratic and Cubic Differential Equation

Question

I came across the following Differential Equation $$y'(x) = a \cdot y(x) + b \cdot y(x)^2 + c \cdot y(x)^3$$ Where $a,b,c$ are numerical coefficients.

I can solve it numerically, but I would like to solve it analytically. Is there some trick or substitution that might make it easier to solve?

Answer 1

Solve $cy^2 + by + a = 0$. Let the roots be $y_0, y_1$. Now decompose $\dfrac{y'}{y(y-y_0)(y-y_1)}$ into $A\dfrac{y'}{y} + B\dfrac{y'}{y - y_0} + C\dfrac{y'}{y - y_1}$. Each of them is easily integratabtle. Can you continue?