PlanetPhysics/Riccati Equation 2

The nonlinear [[../DifferentialEquations/|differential equation]]

$\begin{array}{c}\frac{dy}{dx}=f(x)+g(x)y+h(x)y^2\end{array}$

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

$\begin{array}{c}y:=y_0(x)+\frac{1}{w(x)}\end{array}$

converts (1) to

$\begin{array}{c}\frac{dw}{dx}+[g(x)+2h(x)y_0(x)]w+h(x)=0,\end{array}$

which is a linear differential equation of first order with respect to the [[../Bijective/|function]] \,$w=w(x)$.\\

Example. \, The Riccati equation

$\begin{array}{c}\frac{dy}{x}=3+3x^{2}y-x{y}^2\end{array}$

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 we can put\, $w=uv$,\, $w^{\prime }=u{v}^{\prime }+u^{\prime }v$,\, writing the equation as

$\begin{array}{c}u\cdot (v^{\prime }-3x^{3}v)+u^{\prime }v=x,\end{array}$

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: $u=C+\int x{e}^{-x^3}dx.$ Thus we have $w=w(x)=uv=e^{x^3}[C+\int x{e}^{-x^3}dx],$ whence the general solution of the Riccati equation (4) is Failed to parse (syntax error): {\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.