Ordinary Differential Equations/Ricatti

The Riccati Equation

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

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

so that the equation becomes

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

which is a Bernoulli equation.