A (general) Riccati differential equation is any first-order ordinary differential equation for a function $y = y(t)$ which is of the form
with $a_0(t)\neq 0$ and $a_2(t)\neq 0$.
Riccati equations are to fractional linear transformations as linear first-order ordinary differential equations are to affine transformations, in the following sense. A linear first-order ordinary differential equation
has (under suitable regularity conditions on $a_0$ and $a_1$) solutions $y(t)$ depending in an affine manner on the initial data:
for functions $f,g$ independent of $y(0)$. More generally, any Riccati equation
has (under suitable regularity conditions, and at least for short times) solutions $y(t)$ depending on the initial data as follows:
for functions $f,g,h,i$ independent of $y(0)$.
Unlike solutions of first-order linear equations, solutions of Riccati equations can blow up in finite time; however, these solutions become defined for all time if we let $y$ take values in the projective line rather than the affine line: that is, $\mathbb{R}\mathrm{P}^1$ or $\mathbb{C}\mathrm{P}^1$ rather than $\mathbb{R}$ or $\mathbb{C}$.
Indeed, the group of fractional linear transformations, the projective general linear group $\mathrm{PGL}(2,\mathbb{R})$ or $\mathrm{PGL}(2,\mathbb{C})$, acts naturally on the projective line, with the group of affine transformations being the subgroup that preserves the affine line.
The group of smooth $\mathrm{PGL}(2)$-valued functions on the real line acts on the space of Riccati equations. First, note that this group acts on $\mathbb{P}^1$-valued functions on the real line as follows:
Then, given any Riccati equation
there is another Riccati equation such that $\tilde{y}$ is a solution of this new Riccati equation iff $y$ is a solution of $\star$.
There is a correspondence between Riccati equations and a wide class of 2nd order linear differential equations, namely Riccati equation for $y$ gives a 2nd order linear ODE for $u$ which satisfies $a_2 u' = u y$, namely
Of course, the coefficients are non-constant.
Unlike the Bernoulli differential equation? which has $a_0(z) = 0$ and $a_2(z) \neq 0$, the Riccati equation can not be solved in quadratures in general, unless a a particular solution $y_1$ is known. If so, then we can write $y = y_1 + u$ and write down the equation for $u$ which appears to be an example of a Bernoulli differential equation, hence solvable in quadratures. Regarding that $w = 1/u$ is the substitution for reducing Bernoulli to first order linear ODE, we can reduce the original Riccati to such ODE by substituting $y = y_1 + 1/w$.
For a generalization, the matrix Riccati equation, see the eom article.
Springer eom Riccati equation
Wikipedia: Riccati equation
Last revised on September 7, 2020 at 16:54:36. See the history of this page for a list of all contributions to it.