Ordinary Differential Equations/Riccati

The Riccati Equation

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

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

$\frac{d{y}_1}{dx}+\frac{dz}{dx}+f(x)(y_1^2+2y_{1}z+z^2)+g(x)(y_1+z)+h(x)$
$=\frac{dz}{dx}+(2y_{1}f(x)+g(x))z+f(x)z^2=0$

which is a Bernoulli equation.