Equation of Riccati

In Mathematical, a equation of Riccati is a ordinary differential equation form

$y\text{'}\; =\; q\_0\; (X)\; +\; q\_1\; (X)\; there\; +\; q\_2\; (X)\; y^2\; \backslash ,$.

Where $q\_0\; \backslash ,$, $q\_1\; \backslash ,$ and $q\_2\; \backslash ,$ are three functions, often selected continuous with actual values on a common sometimes interval but one meets them with complex values.

It bears this name in the honor of Jacopo Francesco Riccati (1676-1754) and of its son Vincenzo Riccati (1707-1775).

There does not exist, in general, of resolution by squaring with such an equation but, there exists a method of resolution as soon as one knows a particular solution of it.

Historical aspect

In 1720, Francesco Riccati present at his/her friend, Giovanni Rizzetti, two differential equations that he seeks to solve

• $y\text{'}\; =\; ay^2+bx\; +\; cx^2\; \backslash ,$ where has , B and C are real constants (1)
• $y\text{'}\; =\; ay^2\; +\; bx^m\; \backslash ,$ where has , B and m is real constants (2)

The first equation is resulting from the study of a plane movement checking the following linear differential equation

$$

\begin{pmatrix} x' \ \ y' \ \ \ end {pmatrix} = \ begin {pmatrix} a&b \ \ c&d \ \ \end{pmatrix}\begin{pmatrix} X \ \ there \ \ \end{pmatrix} where X and is there the punctual coordinates M moving. By being interested in the slope Z line ( OM ), it proves that Z must check an equation of the type (1), from where its desire studied the general solutions of them.

The second equation, was solved only partially by its author and the Bernoulli (Nicolas 1 {{er}} and Daniel particularly). His/her son, Vicenzo Riccati, developed of it a method of resolution by Tractoire. Goldbach was harnessed there also and more recently Liouville which proved (1841) that apart from the case

$m\; =\; \backslash \; frac\; \{(-\; 4\; H)\}\{(2\; H\; \backslash \; pm\; 1)\}$ where H is a natural entirety,

the equation is not resolvable by squarings.

The equations of Riccati spread then with any equation of the form

$y\text{'}\; =\; q\_0\; (X)\; +\; q\_1\; (X)\; there\; +\; q\_2\; (X)\; y^2\; \backslash ,$.

For certain conditions on $q\_0\; \backslash ,$, $q\_1\; \backslash ,$, $q\_2\; \backslash ,$, the equation is resolvable by squaring. Thanks to the Theorem of Cauchy-Lipschitz, one proves that, if $q\_0\; \backslash ,$, $q\_1\; \backslash ,$, $q\_2\; \backslash ,$ are continuous functions, then it exists solutions with the equation of Riccati. Finally one shows that, if one knows a particular solution of it, an equation of Riccati is brought back by change of variable, with a equation of Bernoulli.

Resolution knowing a particular solution

If it is possible to find a solution $y\_1$, then the general solution is form

$there\; =\; y\_1\; +\; U\; \backslash ,$

By replacing

$there\; \backslash ,$ by $y\_1\; +\; U\; \backslash ,$

in the equation of Riccati, one obtains:

$y\_1\text{'}\; +\; u\text{'}\; =\; q\_0\; +\; q\_1\; (y\_1\; +\; U)\; +\; q\_2\; (y\_1\; +\; U)\; ^2\; \backslash ,$

and like

$y\_1\text{'}\; =\; q\_0\; +\; q\_1\; y\_1\; +\; q\_2\; y\_1^2\; \backslash ,$

one a:

$u\text{'}\; =\; q\_1\; U\; +\; 2\; q\_2\; y\_1\; U\; +\; q\_2\; u^2\; \backslash .$

However

$u\text{'}\; -\; (q\_1\; +\; 2\; q\_2\; y\_1)\; U\; =\; q\_2\; u^2\; \backslash ,$

who is a equation of Bernoulli. Substitution necessary to the solution of this equation of Bernoulli is then:

$Z\; =\; u^\; \{1-2\}\; =\; \backslash \; frac\; \{1\}\; \{U\}$

To substitute

$there\; =\; y\_1\; +\; \backslash \; frac\; \{1\}\; \{Z\}$

directly in the equation of Riccati the linear equation gives:

$z\text{'}\; +\; (q\_1\; +\; 2\; q\_2\; y\_1)\; Z\; =\; -\; q\_2\; \backslash ,$

The general solution of the equation of Riccati is then given by:

$there\; =\; y\_1\; +\; \backslash \; frac\; \{1\}\; \{Z\}$

where Z is the general solution of the linear equation quoted above.

Fields of use

One meets equations of Riccati in Quantum physics in problems relating to the equation of Schrödinger, in the equation of the waves, in quadratic linear optimal order, or even in the equation of the propagation of heat in sinusoidal Régime. In these cases, the function $q\_1$ is a function with complex values.

One also meets them in financial Mathématiques in the problems relating to the modeling of interest rates. More particularly in the study of the Model Cox-Ingersoll-Ross.

References

• equation of Riccati on the old site of Serge Mehl: ChronoMath .
• history of the equations of Riccati
• equation of riccati resolution by squaring by Rene Lagrange
• Resolution by tractive
• Gallica, supplement of the encyclopedia of Diderot (p 648 for the equation of Riccati)

----

In famous the Encyclopedia, Dictionary reasoned of Sciences, Arts and Trades published under the direction of Diderot, Alembert written:

RICCATI (equation of) Algebra. Integral calculus. One thus calls a first order differential equation with two variables which the count Ricati proposed to the geometricians about 1720, & of which nobody still gave general solution. Perhaps it is not likely to have one in finished terms of them.

This equation is form $dy\; +\; y^2\; dx\; +\; has\; x^m\; dx\; =\; 0$…

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