The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 248 Wednesday, February 4, 2004 12:51 PM

248 The Mathematics of Financial Modeling and Investment Management

a()xy()n+ a^1

n– 1 ()y

(n– 1 ) ()y()+ a

n x + ...+ a 1 x 0 ()xybx+ ()=^0

If the function bis identically zero, the equation is said to be homoge-

neous.

In cases where the coefficients a’s are constant, Laplace transforms

provide a powerful method for solving linear differential equation. Con-

sider, without loss of generality, the following linear equation with con-

stant coefficients:

n

()+ a

n– 1

a y n y (n–^1 ) + ...+ a ()^1 ()

1 y + a 0 y= bx

together with the initial conditions: y(0) = y 0 ,...,y(n–1)(0) = y( 0 n–1). In cases in

which the initial point is not the origin, by a variable transformation we

can shift the origin.

Let’s recall the formula to Laplace-transform derivatives presented

in Chapter 4. For one-sided Laplace transforms the following formulas

hold:

L

df x

--------------= sL[fx ()

 dx 

()

()]– f 0

n

L

d fx

0 ... f

(n– 1 )

---------------- = sL[fx ()

 dxn 

() n

()]– s

n– 1

f' ()– – 0

Suppose that a function y= y(x) satisfies the previous linear equation

with constant coefficients and that it admits a Laplace transform. Apply

one-sided Laplace-transform to both sides of the equation. If Y(s) =

L[y(x)], the following relationships hold:

La( ny()n + an– 1 y(n–^1 )+ ...+ a 1 y()^1 + a 0 y)= Lbx[ ()]

()

a[s 0

n

Ys

1

()– s 0

n– 1

y ()– ...– y

(n– 1 )

n ()]

+ an– 1 [sn–^1 Ys()– sn–^2 y ()^1 () 0 – ...– y(n–^2 )() 0 ]

+ ...+ a 0 Ys()= Bs()