Kovacic's algorithm Is there any reference with some example, about how to solve a "riccati" equation in this (below) form :$$y'(x)+a(x)y^2(x)+b(x)y(x)+c(x)=0$$ by Kovacic's algorithm?
Or can anybody help me to understand how Kovacic's algorithm works?
(In Kovacic's 1985 article, the algorithm is described only for the case of $y''=ry$ with some complicated examples. I am searching for some easier examples. I am new to this study.)
Thanks in advance.
 A: In order to use Kovacic's result, we shall transform your equation into precisely the one he studies. All the coefficients will be functions of $x$, so for an easier typing I shall omit the argument. Keep in mind that Kovacic's algorithm applies to equations the coefficients of which are rational functions of a complex variable.
1) First, note that your equation is not linear, so Kovacic's algorithm cannot be applied to it. Fortunately, if $a \ne 0$, the usual change of unknown $w = -a y$ transforms your equation into another Riccati equation:
$$ w' = w^2 + (\frac {a'} a - b) w + a c .$$
2) The above is still not linear, so we next perform the usual change of unknown $w = - \frac {v'} v$, obtaining
$$v'' + (b - \frac {a'} a) v' + ac = 0 .$$
3) Let $f = b - \frac {a'} a$. In order to get rid of the term containing $v'$, make the last substitution $v = \Bbb e ^{\frac f 2} u$, obtaining
$$u'' = -(\frac {(f')^2} 4 + ac) u .$$
Let $r = -(\frac {(f')^2} 4 + ac) = \frac {a'' a - a^2 b' - (a')^2 - 4 a^3 c} {4 a^2}$ in order for your equation to become $u'' = r u$.
Now you may apply Kovacic's algorithm if $r$ is a fraction of polynomial functions of a complex variable. Unfortunately, there is nothing to be done in the general case if you do not provide the concrete expressions of $a,b,c$.
