Non-linear first order ODE

Question

I need help with the following riccati equation

$$\ y' = Ay^2 + By + C $$

in the situation where discriminant < 0

It must be solved analytically with steps.

Thanks

Answer 1

HINT:

$$y'(x)=ay(x)^2+by(x)+c\Longleftrightarrow$$ $$\frac{\text{d}y(x)}{\text{d}x}=ay(x)^2+by(x)+c\Longleftrightarrow$$ $$\frac{\frac{\text{d}y(x)}{\text{d}x}}{ay(x)^2+by(x)+c}=1\Longleftrightarrow$$ $$\int\frac{\frac{\text{d}y(x)}{\text{d}x}}{ay(x)^2+by(x)+c}\space\text{d}x=\int 1\space\text{d}x\Longleftrightarrow$$ $$\frac{2\arctan\left(\frac{2ay(x)+b}{\sqrt{4ac-b^2}}\right)}{\sqrt{4ac-b^2}}=x+k$$

With $k$ is an arbitrary constant