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476 Chapter 11 Systems of Ordinary Differential Equations

a 5. Using the Runge-Kutta procedure, solve the following initial-value problem on the in-
terval 0 t 2π. Plot the resulting curves (x 1 (t), x 2 (t)) and (x 3 (t), x 4 (t)). They should
be circles.
⎧ ⎡ ⎤
x 3
⎪
⎪
⎪
⎪ ⎢ x 4 ⎥
⎪
⎨ ⎢ −3/2 ⎥
−x 1 x + x
X = ⎢ 2 2 ⎥
⎣ 1 2 ⎦
⎪ −3/2
⎪ 2 2
⎪ −x 2 x + x
⎪ 1 2
⎪
⎩ T
X(0) = [1, 0, 0, 1]
6. Solve the problem
⎧

x = 1
⎪ 0
⎪
⎪
⎨
x =−x 2 + cos x 0
1

⎪ x = x 1 + sin x 0
⎪ 2
⎪
⎩
x 0 (1) = 1 x 1 (1) = 0 x 2 (1) =−1
Use the Runge-Kutta method and the interval −1 t 2.
a
7. Write and test a program, using the Taylor series method of order 5, to solve the system
⎧
2
⎪ x = tx − y + 3t

⎨
2
y = x − ty − t 2

⎪
⎩
x(5) = 2 y(5) = 3
−3
on the interval [5, 6] using h = 10 . Print values of x and y at steps of 0.1.
8. Print a table of sin t and cos t on the interval [0,π/2] by numerically solving the system
⎧
⎪ x = y

⎨
y =−x

⎪
x(0) = 0 y(0) = 1
⎩
9. Write a program for using the Taylor series method of order 3 to solve the system
⎧ 2

x = tx + y − t

⎪
⎪
⎪
y = ty + 3t
⎨

⎪ z = tz − y + 6t 3

⎪
⎪
⎩
x(0) = 1 y(0) = 2 z(0) = 3
on the interval [0, 0.75] using h = 0.01.
10. Write and test a short program for solving the system of differential equations
⎧ 3 2 2
⎪ y = x − t y − t

⎨
2
4
x = tx − y + 3t

⎪
⎩
y(2) = 5 x(2) = 3
over the interval [2, 5] with h = 0.25. Use the Taylor series method of order 4.
11. Recode and test procedure RK4 System2 using a computer language that supports
vector operations.