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474 Chapter 11 Systems of Ordinary Differential Equations

(3) The Runge-Kutta method of order 4 is
h
X(t + h) = X + (K 1 + 2K 2 + 2K 3 + K 4 )
6
where
⎧
K 1 = F(t, X)
⎪
⎪
⎪ 1 1
⎨
K 2 = F t + h, X + hK 1
2 2

1 1
⎪ K 3 = F t + h, X + hK 2
⎪ 2 2
⎪
⎩
K 4 = F(t + h, X + hK 3 )
Here, X = X(t), and all quantities are vectors with n components except variables t and h.
(4) We can absorb the t variable into the vector by letting x 0 = t and then writing the
autonomous form for the system of ordinary differential equations in vector notation as

X = F(X)

X(a) = S, given
where vectors are defined to have n + 1 components. Then
⎧ T
X = [x 0 , x 1 , x 2 ,..., x n ]
⎪
⎪
⎪
X = [x , x , x ,..., x ]
⎨ T
0 1 2 n
F = [1, f 1 , f 2 ,..., f n ] T
⎪
⎪
⎪
⎩ T
X(a) = [a, x 1 (a), x 2 (a),..., x n (a)]
(5) The Runge-Kutta method of order 4 for the system of ordinary differential equations
in autonomous form is
h
X(t + h) = X + (K 1 + 2K 2 + 2K 3 + K 4 )
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 F and K i are vectors with n + 1 components except the
variables t and h.
Problems 11.1
a 1. Consider

x = y x(0) =−1

with
y = x y(0) = 0

Write down the equations, without derivatives, to be used in the Taylor series method
of order 5.