Riccati Equation - Obtaining Solutions By Quadrature

Obtaining Solutions By Quadrature

The correspondence between Riccati equations and second-order linear ODEs has other consequences. For example, if one solution of a 2nd order ODE is known, then it is known that another solution can be obtained by quadrature, i.e., a simple integration. The same holds true for the Riccati equation. In fact, if one particular solution can be found, the general solution is obtained as

Substituting

in the Riccati equation yields

and since

or

which is a Bernoulli equation. The substitution that is needed to solve this Bernoulli equation is

Substituting

directly into the Riccati equation yields the linear equation

A set of solutions to the Riccati equation is then given by

where z is the general solution to the aforementioned linear equation.