Riccati equation

The nonlinear differential equation

$\displaystyle \frac{dy}{dx} = f(x)+g(x)y+h(x)y^2$ (1)

is called the Riccati equation. If $ h(x) \equiv 0$ , it becomes a linear differential equation; if $ f(x) \equiv 0$ , then it becomes a Bernoulli equation. There is no general method for integrating explicitely the equation (1), but via the substitution

$\displaystyle y \,:=\, -\frac{w'(x)}{h(x)w(x)}$

one can convert it to a second order homogeneous linear differential equation with non-constant coefficients.

If one can find a particular solution $ y_0(x)$ , then one can easily verify that the substitution

$\displaystyle y \,:=\, y_0(x)+\frac{1}{w(x)}$ (2)

converts (1) to

$\displaystyle \frac{dw}{dx}+[g(x)\!+\!2h(x)y_0(x)]\,w+h(x) = 0,$ (3)

which is a linear differential equation of first order with respect to the function $ w =w(x)$ .

Example. The Riccati equation

$\displaystyle \frac{dy}{x} = 3+3x^2y-xy^2$ (4)

has the particular solution $ y := 3x$ . Solve the equation.

We substitute $ y := 3x+\frac{1}{w(x)}$ to (4), getting

$\displaystyle \frac{dw}{dx}-3x^2w-x = 0.$

For solving this first order equation we can put $ w = uv$ , $ w' = uv'+u'v$ , writing the equation as

$\displaystyle u\cdot(v'-3x^3v)+u'v = x,$ (5)

where we choose the value of the expression in parentheses equal to 0:

$\displaystyle \frac{dv}{dx}-3x^2v = 0$

After separation of variables and integrating, we obtain from here a solution $ v = e^{x^3}$ , which is set to the equation (5):

$\displaystyle \frac{du}{dx}e^{x^3} = x$

Separating the variables yields

$\displaystyle du = \frac{x}{e^{x^3}}\,dx$

and integrating:

$\displaystyle u = C+\int xe^{-x^3}\,dx.$

Thus we have

$\displaystyle w = w(x) = uv = e^{x^3}\left[C+\int xe^{-x^3}\,dx\right],$

whence the general solution of the Riccati equation (4) is

$\displaystyle \displaystyle y \,:=\, 3x+\frac{e^{-x^3}}{C+\int xe^{-x^3}\,dx}.\\ $

It can be proved that if one knows three different solutions of Riccati equation (1), then any other solution may be expressed as a rational function of the three known solutions.



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