Riccati equation

The nonlinear differential 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), butvia 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

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

converts (1) to

${\displaystyle \frac{dw}{dx}}+[g(x)+2h(x)y_0(x)]w+h(x)=\mathrm{\hspace{0.33em}0},$(3)

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

Example.  The Riccati equation

${\displaystyle \frac{dy}{x}}=\mathrm{\hspace{0.33em}3}+3x^{2}y-x{y}^2$(4)

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

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

$$\frac{dw}{dx}-3x^{2}w-x=\mathrm{\hspace{0.33em}0}.$$

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

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

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

$$\frac{dv}{dx}-3x^{2}v=\mathrm{\hspace{0.33em}0}$$

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:

$$u=C+\int x{e}^{-x^3}𝑑x.$$

Thus we have

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

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

$$y=\mathrm{\hspace{0.33em}3}x+\frac{e^{-x^3}}{C+\int x{e}^{-x^3}𝑑x}.$$

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.