Ordinary Differential Equations/Riccati

The Riccati Equation

${dy \over dx}+f(x)y^{2}+g(x)y+h(x)=0$

is different from the previous differential equations because, in general, the solution is not expressible in terms of elementary integrals.

However, we can obtain a general solution from a single particular solution when one is known.

Let $y_{1}(x)$ be a particular solution, and let $y(x)=y_{1}+z$

so that the equation becomes

${dy_{1} \over dx}+{dz \over dx}+f(x)(y_{1}^{2}+2y_{1}z+z^{2})+g(x)(y_{1}+z)+h(x)$
$={dz \over dx}+(2y_{1}f(x)+g(x))z+f(x)z^{2}=0$

which is a Bernoulli equation.