The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 241 Wednesday, February 4, 2004 12:51 PM

Differential Equations and Difference Equations 241

Order and Degree of an ODE

A differential equation is classified in terms of its order and its degree.

The order of a differential equation is the order of the highest derivative

in the equation. For example, the above differential equation is of order n

since the highest order derivative is Y(n)(x).The degree of a differential

equation is determined by looking at the highest derivative in the differen-

tial equation. The degree is the power to which that derivative is raised.

For example, the following ordinary differential equations are first

degree differential equations of different orders:

Y(1)(x) – 10Y(x) + 40 = 0 (order 1)

4 Y(3)(x) + Y(2)(x) + Y(1)(x) – 0.5Y(x) + 100 = 0 (order 3)

The following ordinary differential equations are of order 3 and fifth

degree:

4 [Y(3)(x)]^5 + [Y(2)(x)]^2 + Y(1)(x) – 0.5Y(x) + 100 = 0

4 [Y(3)(x)]^5 + [Y(2)(x)]^3 + Y(1)(x) – 0.5Y(x) + 100 = 0

When an ordinary differential equation is of the first degree, it is said to

be a linear ordinary differential equation.

Solution to an ODE

Let’s return to the general ODE. A solution of this equation is any function

y(x) such that:

n

Fxyx

1

(), ...,y

()

[ , (),y x

()

x ()]= 0

In general there will be not one but an infinite family of solutions. For

example, the equation

Y()^1 ()x = αYx()

admits, as a solution, all the functions of the form

yx()= Cexp(αx)

To identify one specific solution among the possible infinite solu-

tions that satisfy a differential equation, additional restrictions must be