The non-linear equat

where the coefficients p(t), b(t), q(t) are functions of the real variable t, is known as the Riccati equation. The solution of (C.1) is known only for some special forms of the coefficients in it.

By transforming to the variable x(t) defined by

the Riccati equation transforms to the second-order ordinary linear differential equation for x(t):

The solution of the Riccati equation may thus be used to solve (C.3).

A solvable case of (C.1) arises when the coefficients in it are independent of t. Assuming that to be the case, rewrite (C.1) as

Substitute (C.5) in (C.4) and integrate the resulting equation to obtain

On rearranging the terms, the expression for y(t) reads

The same result can be arrived at by solving the linear equation (C.3) corresponding to (C.1) (see Ex. C.2)

Ex. C.1. Show that even though x(t) in (C.2) is the linear combination of two linearly independent solutions x1(t) and x2(t) of (C.3), y(t) is determined only by its initial value.

Ex. C.2. Assuming the coefficients in (C.1) to be independent of t, solve the associated linear equation (C.3) for x(t) and show that the solution y(t) so obtained is the same as (C.8).