PlanetPhysics/Riccati Equation 2

The nonlinear differential equation $$\begin{matrix} \frac{dy}{dx} = f(x)+g(x)y+h(x)y^2 \end{matrix}$$ 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 $$y \,:=\, -\frac{w'(x)}{h(x)w(x)}$$ 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 $$\begin{matrix} y \,:=\, y_0(x)+\frac{1}{w(x)} \end{matrix}$$ converts (1) to $$\begin{matrix} \frac{dw}{dx}+[g(x)\!+\!2h(x)y_0(x)]\,w+h(x) = 0, \end{matrix}$$ which is a linear differential equation of first order with respect to the function \,$$w =w(x)$$.\\

Example. \, The Riccati equation $$\begin{matrix} \frac{dy}{x} = 3+3x^2y-xy^2 \end{matrix}$$ has the particular solution\, $$y := 3x$$.\, Solve the equation.

We substitute\, $$y := 3x+\frac{1}{w(x)}$$\, to (4), getting $$\frac{dw}{dx}-3x^2w-x = 0.$$ For solving this first order equation we can put\, $$w = uv$$,\, $$w' = uv'+u'v$$,\, writing the equation as $$\begin{matrix} u\cdot(v'-3x^3v)+u'v = x, \end{matrix}$$ where we choose the value of the expression in parentheses equal to 0: $$\frac{dv}{dx}-3x^2v = 0$$ After separation of variables and integrating, we obtain from here a solution\, $$v = e^{x^3}$$,\, which is set to the equation (5): $$\frac{du}{dx}e^{x^3} = x$$ 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 $$ y \,:=\, 3x+\frac{e^{-x^3}}{C+\int xe^{-x^3}\,dx}.\\$$

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.