The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 244 Wednesday, February 4, 2004 12:51 PM

244 The Mathematics of Financial Modeling and Investment Management

Solving this system means finding a set of functions y 1 ,...,yn that satisfy

the system as well as the initial conditions:

y 1 ()x 0 = y 10 ,, ...yn()x 0 = yn 0

Systems of orders higher than one can be reduced to first-order systems

in a straightforward way by adding new variables defined as the deriva-

tives of existing variables. As a consequence, an n-th order differential

equation can be transformed into a first-order system of n equations.

Conversely, a system of first-order differential equations is equivalent to

a single n-th order equation.

To illustrate this point, let’s differentiate the first equation to obtain

d

2

y 1 ∂f 1 ∂f 1 dy 1 ∂f 1 dyn

------------ = --------+ --------- ---------+ ...+ --------- ---------

dx^2 ∂x ∂y^1 dx ∂yn dx

Replacing the derivatives

dy 1 dyn

---------,, ... ---------

dx dx

with their expressions f 1 ,...,fn from the system’s equations, we obtain

d^2 y 1

------------ = F 2 (xy,,, 1 ...yn)

dx

2

If we now reiterate this process, we arrive at the n-th order equation:

d()ny

1

--------------- = F(xy,,, 1 ...yn)

() n

dx

n

We can thus write the following system: