is called Riccati equation. If , it is a question of a linear differential equation; if , of a Bernoulli equation. There is no general method for integrating explicitely the equation (1), but via the substitution
one can convert it to a homogeneous linear differential equation with non-constant coefficients.
If one can find a particular solution , then one can easily verify that the substitution
converts (1) to
which is a linear differential equation of first order with respect to the function .
Example. The Riccati equation
has the particular solution . Solve the equation.
We substitute to (4), getting
For solving this first order equation (http://planetmath.org/LinearDifferentialEquationOfFirstOrder) we can put , , writing the equation as
where we choose the value of the expression in parentheses equal to 0:
After separation of variables and integrating, we obtain from here a solution , which is set to the equation (5):
Separating the variables yields
Thus we have
whence the general solution of the Riccati equation (4) is
It may be proved that if one knows three different solutions of Riccati equation (1), the each other solution may be expresses as a rational function of them.
|Date of creation||2013-03-22 18:05:43|
|Last modified on||2013-03-22 18:05:43|
|Last modified by||pahio (2872)|
|Synonym||Riccati differential equation|