Differential equation from physics

Question

How can I solve this differential equation ? $$y'= ay^2+b$$ ($a$ and $b$ are from $ \mathbb Q$ )

The actual form of this formula was $$g =\frac{k}{m}\cdot v^2(t)+ a(t)$$ (from a physics problem).

Answer 1

This is separable.

$$\frac{y'}{y^2+c}=a.$$

The antiderivative depends on the sign of $c$.

$$\frac{y'}{y^2+d^2}=a\to \frac 1d\arctan\frac yd=at+e.$$

$$\frac{y'}{y^2-d^2}=a\to-\frac 1d\text{artanh}\frac yd=at+e.$$