Consider a differential equation

called a *Riccati equation*. Prove that if is a solution of this equation then there are additional solutions

where satisfies a first-order linear differential equation.

*Proof.* Let be a function satisfying

Further, let where satisfies a first-order linear differential equation. Then,

Therefore,

Thus, if

then is a solution of our equation. But,

This is a first-order linear differential equation (since is a function of ). Hence, is a solution of

where is the solution of the first-order linear differential equation