Riccati equation

It is a nonlinear first order ODE that was developed in the context of hydrodynamics. It is of the form

where $P(x),Q(x),R(x)$ are known (sufficiently differentiable) functions. The typical solution method follows these steps:

1. Find a particular solution $y_1$, often by trial and error.
2. Substitute $y=y_1+u$ into the Riccati equation, which transforms it into a Bernoulli equation in terms of $u$. Solve this equation to obtain a one-parameter family of solutions.
3. The general solution is then given by $y=y_1+u$.

In the particular case of the Riccati equation

where $a,b,$ and $\alpha$ are real parameters the function

is a solution of the equation.

Relationship with the Schrödinger Equation

The Riccati equation is intimately linked to second-order linear ordinary differential equations, such as the Schrödinger equation. Consider the Schrödinger-type equation:

If we define $y(x)=\frac{u^{\prime }(x)}{u(x)}$, then $y(x)$ satisfies the Riccati equation:

Conversely, given a solution $y(x)$ of the Riccati equation $y^{\prime }+y^2=g(x)$, we can obtain a family of solutions to the Schrödinger equation via:

where $C$ is an arbitrary constant.

Geometric Interpretation

The set of solutions to the Schrödinger equation forms a 2-dimensional vector space (the "plane of solutions"). Since a single solution $y(x)$ of the Riccati equation determines $u(x)$ only up to a constant scaling factor $C$, it corresponds to a line (a 1-dimensional subspace) through the origin in this plane of solutions.

The general solution of the Riccati equation is a one-parameter family of functions. This family corresponds to the set of all lines in the plane of solutions of the Schrödinger equation, which is topologically a projective line ${\mathbb{P}}^1$.

More on the relation: A Riccati-type equation can be transformed into a second order linear homogeneous differential equation (see this). Moreover, it can be "approximately transformed" into a Schrodinger equation (see this).