# Riccati equation

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

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

 $y\;:=\;-\frac{w^{\prime}(x)}{h(x)w(x)}$

one can convert it to a 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

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

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

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

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

For solving this first order equation (http://planetmath.org/LinearDifferentialEquationOfFirstOrder) we can put  $w=uv$,  $w^{\prime}=uv^{\prime}+u^{\prime}v$,  writing the equation as

 $\displaystyle u\cdot(v^{\prime}-3x^{3}v)+u^{\prime}v\;:=\;x,$ (5)

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

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

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

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

Separating the variables yields

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

and integrating:

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

Thus we have

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

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

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

It may be proved that if one knows three different solutions of Riccati equation (1), the each other solution may be expresses as a rational function of them.

Title Riccati equation RiccatiEquation 2013-03-22 18:05:43 2013-03-22 18:05:43 pahio (2872) pahio (2872) 12 pahio (2872) Result msc 34A34 msc 34A05 Riccati differential equation BernoulliEquation