3.3.21.1 Direct solution of Riccati
There is no general method to solve the general Riccati ode. These are special cases to
try
3.3.21.1.1 Case 1
If \(f_{0},f_{1},f_{2}\) are constants then this is separable ode and can easily be solved.
3.3.21.1.2 Case 2 (particular solution is known)
Assume we can find a particular solution \(y_{1}\) to the general Riccati ode \(y^{\prime }=f_{0}\left ( x\right ) +f_{1}\left ( x\right ) y+f_{2}\left ( x\right ) y^{2}\). Then let \(y=y_{1}+u\). The Riccati
ode becomes a Bernoulli ode.
\begin{align*} \left ( y_{1}+u\right ) ^{\prime } & =f_{0}+f_{1}\left ( y_{1}+u\right ) +f_{2}\left ( y_{1}+u\right ) ^{2}\\ y_{1}^{\prime }+u^{\prime } & =f_{0}+f_{1}y_{1}+f_{1}u+f_{2}\left ( y_{1}^{2}+u^{2}+2y_{1}u\right ) \\ y_{1}^{\prime }+u^{\prime } & =f_{0}+f_{1}y_{1}+f_{1}u+f_{2}y_{1}^{2}+f_{2}u^{2}+2f_{2}y_{1}u\\ y_{1}^{\prime }+u^{\prime } & =\overbrace {f_{0}+f_{1}y_{1}+f_{2}y_{1}^{2}}+f_{1}u+f_{2}u^{2}+2f_{2}y_{1}u\\ u^{\prime } & =f_{1}u+f_{2}u^{2}+2f_{2}y_{1}u\\ & =u\left ( f_{1}+2f_{2}y_{1}\right ) +f_{2}u^{2}\end{align*}
Which is Bernoulli ode. But this assumes we are able to find particular solution \(y_{1}\) to
the general Riccati ode. There is no method to do that. So this case will not be
tried.
3.3.21.1.3 References used
- https://mathworld.wolfram.com/RiccatiDifferentialEquation.html
- https://math24.net/riccati-equation.html
- https://encyclopediaofmath.org/wiki/Riccati_equation
- https://www.youtube.com/watch?v=iuHDmZ8VutM
- paper: Methods of Solution of the Riccati Differential Equation. By D. Robert
Haaheim and F. Max Stein. 1969