There is no general method to solve the general Riccati ode. These are special cases to try
2.2.17.1.1 Case 1 If \(f_{0},f_{1},f_{2}\) are constants then this is separable ode and can easily be solved.
2.2.17.1.2 Case 2 If \(f_{1}=0\) then we have the reduced Riccati \[ y^{\prime }=f_{0}\left ( x\right ) +f_{2}\left ( x\right ) y^{2}\] For the case of \(f_{0}=cx^{n}\) where \(c_{1}\) is constant and \(f_{2}=c_{2}\) is also a constant, then the above becomes\[ y^{\prime }=c_{1}x^{n}+c_{2}y^{2}\] Now it depends on \(n\). The case of \(n=-2\) is \(y^{\prime }=\frac {c_{1}}{x^{2}}+c_{2}y^{2}\) can be solved using the substitution \(y=\frac {1}{u}\). Hence \(y^{\prime }=-\frac {u^{\prime }}{u^{2}}\) and the ode becomes\begin {align*} -\frac {u^{\prime }}{u^{2}} & =\frac {c_{1}}{x^{2}}+c_{2}\frac {1}{u^{2}}\\ -u^{\prime } & =c_{1}\frac {u^{2}}{x^{2}}+c_{2}\\ u^{\prime } & =-c_{1}\frac {u^{2}}{x^{2}}-c_{2} \end {align*}
Which is first order Homogeneous ode type (see earlier section).
The case of \(n=-4k\left ( 2k-1\right ) \) where \(k=0,\pm 1,\pm 2,\cdots \) are all solvable by algebraic, exponential and logarithmic function. For all other values, Liouville proved no solution exist in terms of elementary functions. These \(n\) values come out to be \(n=\left \{ \cdots ,-112,-60,-24,-4,0,-12,-40,-84,\cdots \right \} \). For example for \(n=-4\)\[ y^{\prime }=\frac {c_{1}}{x^{4}}+c_{2}y^{2}\] This is solved by converting to second order ode using \(y=\frac {-u^{\prime }}{c_{2}u}\) which result in ode which can be solved as Bessel ode. Similarly for all other \(n\) values listed above. I need to look into this. When I tried \(n=-3\) I also got solution in terms of Bessel functions. So what is the difference?
2.2.17.1.3 Case 3 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.