A (general) Riccati differential equation is any first-order ordinary differential equation for a function which is of the form
with and .
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 and ) solutions depending in an affine manner on the initial data:
for functions independent of . More generally, any Riccati equation
has (under suitable regularity conditions, and at least for short times) solutions depending on the initial data as follows:
for functions independent of .
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 take values in the projective line rather than the affine line: that is, or rather than or .
Indeed, the group of fractional linear transformations, the projective general linear group or , acts naturally on the projective line, with the group of affine transformations being the subgroup that preserves the affine line.
The group of smooth -valued functions on the real line acts on the space of Riccati equations. First, note that this group acts on -valued functions on the real line as follows:
Then, given any Riccati equation
there is another Riccati equation such that is a solution of this new Riccati equation iff is a solution of .
There is a correspondence between Riccati equations and a wide class of 2nd order linear differential equations, namely Riccati equation for gives a 2nd order linear ODE for which satisfies , namely
Of course, the coefficients are non-constant.
Unlike the Bernoulli differential equation? which has and , the Riccati equation can not be solved in quadratures in general, unless a a particular solution is known. If so, then we can write and write down the equation for which appears to be an example of a Bernoulli differential equation, hence solvable in quadratures. Regarding that is the substitution for reducing Bernoulli to first order linear ODE, we can reduce the original Riccati to such ODE by substituting .
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.