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

$${dy_1 \over dx} + {dz \over dx} + f(x)(y_1^2+2y_1z+z^2)+g(x)(y_1+z)+h(x)$$

$$= {dz \over dx} + (2y_1f(x)+g(x))z+f(x)z^2=0$$

which is a Bernoulli equation.