Ordinary Differential Equations/Ricatti

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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.