Riccati Equations

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.
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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.
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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.
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This verifies the solution.
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