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The Solution of the Riccati Equation

In any mathematical investigation of the dynamics of a system, the introduction of a
nonlinearity leads to some form of the Riccati equation:

dy/dt = a(t) + b(t)y + c(t)y^{2}

Because the Riccati equation does not lend itself to solution by the techniques that
produce solutions to linear differential equations typically the analysis is stalled.
Moreover because of the similarity of the Riccati equation to the equations analyzed
in Chaos Theory there is a presumption that further analysis will involve the concepts of
Chaos Theory. But there is a line of analysis of the Riccati equation that is relatively
simple.

The first important result in the analysis of the Riccati equation is that if one
solution is known then a whole family of solutions can be found.

Let x(t) be a solution to the Riccati equation and suppose z(t) is a function such
that

y(t) = x(t) + 1/z(t)

is a solution to the Riccati equation.
As is shown in Appendix I this requires that z(t) satisfy the linear differential
equation

dz/dt = -(b(t) + 2c(t)x(t))z - c(t)

This is an equation of the form

dz/dt = A(t)z - B(t)

which has a solution of

z(t) = e^{A(t)}[∫^{t}e^{-A(s)}B(s)ds + λ]

where λ is an arbitrary constant.
Thus the solutions for z(t) can be determined as a one-parameter family of functions and
hence the solutions to the Riccati equation can be expressed as a one-parameter family
of functions.

But it is usually the case that not even one solution of the Riccati equation is
known.

The second important result in the analysis of the solution of the Riccati equation
is that the general equation can be reduced to a simpler form by the substitution

w(t) = exp(∫^{t}c(s)ds)y(t)
which leads to the equation
dw/dt = p(t) + q(t)w^{2}

where p(t)=-a(t)exp(∫^{t}c(s)ds) and q(t)=-b(t)exp(-∫^{t}c(s)ds).

A substitution of -Q(t)w(t)=(dz/dt)/z reduces the previous equation to

d/dt[-(1/q(t))du/dt] - p(t)u(t) = 0

This second order differential equation may lend itself to solution more readily than
the previous forms of the Riccati equation.