Riccati equation

The nonlinear differential equation

dydx=f(x)+g(x)y+h(x)y2 (1)

is called Riccati equation.  If  h(x)0,  it is a question of a linear differential equation; if  f(x)0,  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  y0(x),  then one can easily verify that the substitution

y:=y0(x)+1w(x) (2)

converts (1) to

dwdx+[g(x)+2h(x)y0(x)]w+h(x)= 0, (3)

which is a linear differential equation of first order with respect to the function  w=w(x).

Example.  The Riccati equation

dyx= 3+3x2y-xy2 (4)

has the particular solution  y:=3x.  Solve the equation.

We substitute  y:=3x+1w(x)  to (4), getting

dwdx-3x2w-x= 0.

For solving this first order equation (http://planetmath.org/LinearDifferentialEquationOfFirstOrder) we can put  w=uv,  w=uv+uv,  writing the equation as

u(v-3x3v)+uv:=x, (5)

where we choose the value of the expression in parentheses equal to 0:

dvdx-3x2v= 0

After separation of variablesMathworldPlanetmath and integrating, we obtain from here a solution  v=ex3,  which is set to the equation (5):


Separating the variables yields


and integrating:


Thus we have


whence the general solution of the Riccati equation (4) is

y= 3x+e-x3C+xe-x3𝑑x.

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.

Title Riccati equation
Canonical name RiccatiEquation
Date of creation 2013-03-22 18:05:43
Last modified on 2013-03-22 18:05:43
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Result
Classification msc 34A34
Classification msc 34A05
Synonym Riccati differential equation
Related topic BernoulliEquation