The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 245 Wednesday, February 4, 2004 12:51 PM

Differential Equations and Difference Equations 245

dy 1

--------- = f 1 (xy,,, 1 ...yn)

dx



d

(^2) y
1
------------= F 2 (xy,,, 1 ...yn)
dx^2

.

.
.

 d()ny 1
---------------() = Fn (xy,,,^1 ...yn)
dx n

We can express y 2 ,...,yn as functions of xy,, ,, 1 y′ ...y
(
1
n – 1 )
by solving,
if possible, the system formed with the first n – 1 equations:
1

y 2 = φ 2 (xy,, ,, 1 y′ 1 ...y 1 ( n –^1 ))

,, ,, y
(
1
n – 1 )
y 3 = φ 3 (xy 1 y′ 1 ... )
.

.
 (^).


n ,, ,, y
(
1
n – 1 ))
yn = φ (xy^1 y′^1 ...
Substituting these expressions into the n-th equation of the previous sys-
tem, we arrive at the single equation:
d()ny
1
, ,, y
(
1
n – 1 )
--------------- = Φ(xy′ 1 ... )
()
dx
n
Solving, if possible, this equation, we find the general solution
y 1 = y 1 (xC, ,, 1 ...Cn)
Substituting this expression for y 1 into the previous system, y 2 ,...,yn can
be computed.