Riccati equation
The Riccati differential equation is a nonlinear ordinary differential equation of the first order of the form
It is named after the mathematician Jacopo Francesco Riccati, an Italian Count (1676-1754), who worked intensively with the classification of differential equations and developed methods to reduce the order of equations.
A general integrating the Riccati equation is not possible with the conventional methods.
Same name Riccati differential equation still carry two other types of equations, which are used for various topics of applied mathematics to finance important.
- 2.1 Formulation of the transform set
- 2.2 proof
Transformation in the case of a known solution
Suppose you already have a solution (such as rates) have been found. Then can the Riccati differential equation completely solve, since the discovery of the other solutions are now reduced to a Bernoulli differential equation which can be solved easily.
Formulation of the transform set
There are, as well as a solution of the differential equation riccatischen
And a solution of Bernoulli's differential equation
Then
The solution of the differential equation riccatischen
Evidence
It is
Is trivially satisfied while the initial value.
Forming a linear differential equation of second order
In general, regardless of whether you have found a special solution that allows the Riccati differential equation transform to a linear second order differential equation with non- constant coefficients. If by chance the coefficients to be constant, this equation can be easily transformed completely solve the problem using the method of variation of constants. In the case of non-constant coefficients can be very difficult to solve the linear form of riccatischen differential equation.
Formulation of the transform set
There are, as well as continuously differentiable and a solution of the linear differential equation of second order
With for all. Then
The solution of the differential equation riccatischen
Evidence
For clarity, are not written arguments. Applies after the quotient rule
Is trivially satisfied while the initial value.