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472 Chapter 11 Systems of Ordinary Differential Equations

As a result of the preceding remarks, we sacrifice no generality in considering a system
of n + 1 first-order differential equations written as
⎧

⎪ x = f 0 (x 0 , x 1 , x 2 ,..., x n )
0
⎪
⎪
⎪
⎪ x = f 1 (x 0 , x 1 , x 2 ,..., x n )
⎪
1
⎪
⎪
⎪

⎨ x = f 2 (x 0 , x 1 , x 2 ,..., x n )
2
.
⎪ .
⎪ .
⎪
⎪
⎪
⎪ x = f n (x 0 , x 1 , x 2 ,..., x n )
⎪
⎪ n
⎪
⎩
x 0 (a) = s 0 , x 1 (a) = s 1 , x 2 (a) = s 2 ,..., x n (a) = s n all given
We can write this system in general vector notation as

X = F(X)
(7)
X(a) = S, given
where
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
x 0 x f 0 s 0
0
⎢ ⎥ ⎢ x ⎥ ⎢ ⎥ ⎢ ⎥
x 1 f 1 s 1
⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ x ⎥ ⎢ ⎥ ⎢ ⎥
x 2 f 2 s 2
X = ⎢ ⎥ X = ⎢ 2 ⎥ F = ⎢ ⎥ S = ⎢ ⎥
⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥
. . . .
⎣ . ⎦ ⎣ . ⎦ ⎣ . ⎦ ⎣ . ⎦
x n x n f n s n
A system of differential equations without the t variable explicitly present is said to be
autonomous. The numerical methods that we discuss do not require that x 0 = t or f 0 = 1
or s 0 = a.
For an autonomous system, the classical fourth-order Runge-Kutta method for
System (6) uses these formulas:
h
X(t + h) = X + (K 1 + 2K 2 + 2K 3 + K 4 ) (8)
6
where
⎧
K 1 = F(X)
⎪
⎪
⎪ 1
⎨
K 2 = F X + hK 1
2

1
⎪ K 3 = F X + hK 2
⎪ 2
⎪
⎩
K 4 = F(X + hK 3 )
Here, X = X(t), and all quantities are vectors with n+1 components except the variables h.
In the previous example, the procedure RK4 System1 would need to be modified by
beginning the arrays with 0 rather than 1 and omitting the variable t. (We call it RK4 System2
and leave it as Computer Problem 11.1.4.) Then the calling programs would be as follows:
program Test RK4 System2
real h, t; real array (x i ) 0:n
integer n ← 2, nsteps ← 100
real a ← 0, b ← 1
(x i ) ← (0, 1, 0)
h ← (b − a)/nsteps
call RK4 System2(n, h,(x i ), nsteps)
end program Test RK4 System2