General Solution to Ricatti Differential Equations

Question

I’ve been studying general form Ricatti Differential Equations recently, and I’m confused as to why a general form solution is not possible. What would it mean if a general form solution were possible?

Answer 1

If a special solution $y_1$ of the Ricatti Differential Equation is known, then the general solution is obtained as $y=y_1+u$, where $u$ satisfies a Bernoulli equation.

If a special solution of the Ricatti Differential Equation is not known, then, in general, the general form of the solutions is not possible.

Answer 2

The well known way to solve the Riccati ODE is first to find a particular solution. Then it is easy to transform it to a first order linear ODE. See Eqs.[7-8} in : http://mathworld.wolfram.com/RiccatiDifferentialEquation.html

But this method fails if one cannot find a particular solution, which unfortunately is generally the case (except in textbook exercises).

Many people think that a first order ODE is always easier to solve than a second order ODE. This is a big mistake, especially in the case of Riccati ODE.

A generally much efficient approach consists in transforming the Riccati ODE, which is a non-linear fist order ODE, into a second order linear ODE. A second order linear ODE is often easier to solve than a first order non-linear ODE.

It is easy to transform the Riccati ODE to a second order linear ODE : See Eqs.[5-6] in the above referenced page.

EXAMPLE :

In this question Solve $(x^4+y^4)\,\text{d}x-xy\,\text{d}y=0$ a Riccati equation is involved after some preliminary calculus. One cannot find a particular solution of this Riccati equation (Afterwards it appears that this would require Bessel functions, impossible to guess a-priori). Transforming the Riccati equation into the related second order linear equation leads to a Bessel ODE which solution is known. So, the problem apparently unsolvable on closed form is finally solved in terms of Bessel functions.