A Riccati equation is a first-order equation of the form
This equation was used by Count Riccati of Venice (1676-1754) to help in solving second-order ordinary differential equations.
Solving Riccati equations is considerably more difficult than solving linear ODEs.
Here is a simple Riccati equation for which the solution is available in closed form.
Any Riccati equation can be transformed to a second-order linear ODE. If the latter can be solved explicitly, then a solution for the Riccati equation can be derived.
Here is an example of a Riccati equation and the corresponding second-order ODE, which is Legendre's equation.
Finally, consider the following Riccati equation. It is integrable because the sum of the coefficients of the terms on the right-hand side is 0.
This verifies the solution.