Chapter 2.7, Problem 23P

A differential equation of the form

$\frac{dy}{dt}=A\left(t\right)+B\left(t\right)y+C\left(t\right)y^2$ (i)

is called a Riccati equation. Such equations arise in optimal control theory.

If $y_1$ is a known solution of (i), prove that the substitution $y=y_1+v$ transforms (i) into a Bernoulli equation with $n=2$.

Solve the equation $\frac{dy}{dx}+3ty=4-4t^2+y^2$, after showing that it has $y=4t$ as a particular solution.