Page 11 - DOC-20220324-WA0012.-488-500

11.1 Methods for First-Order Systems 475

a 2. How would you solve this system of differential equations numerically?
⎧
t
2
⎪ x = x + e − t 2

⎨ 1 1
x = x 2 − cos t

2
⎪
x 1 (0) = 0 x 2 (1) = 0
⎩
a
3. How would you solve the initial-value problem
⎧
t
⎪ x (t) = x 1 (t)e + sin t − t 2

⎨ 1
t
2
x (t) = [x 2 (t)] − e + x 2 (t)

2
⎪
x 1 (1) = 2 x 2 (1) = 4
⎩
if a computer program were available to solve an initial-value problem of the form

x = f (t, x) involving a single unknown function x = x(t)?
a
4. Write an equivalent system of first-order differential equations without t appearing on
the right-hand side:
⎧ 2 2

⎪ x = x + log(y) + t
⎨
y
y = e − cos(x) + sin(tx) − (xy) 7

⎪
⎩
x(0) = 1 y(0) = 3
Computer Problems 11.1
a 1. Solve the system of differential equations (1) by using two different methods given in
this section and compare the results with the analytic solution.
a 2. Solve the initial-value problem
⎧ 2
⎪ x = t + x − y

⎨
2
y = t − x + y 2

⎪
x(0) = 3 y(0) = 2
⎩
by means of the Taylor series method using h = 1/128 on the interval [0, 0.38]. Include
terms involving three derivatives in x and y. How accurate are the computed function
values?
3. Write the Runge-Kutta procedure to solve
⎧

⎪ x =−3x 2
⎨ 1
1
x = x 1
2 3
⎪
x 1 (0) = 0 x 2 (0) = 1
⎩
on the interval 0 t 4. Plot the solution.
a
4. Write procedure RK4 System2 and a driver program for solving the ordinary differential
−2
equation system given by Equation (2). Use h =−10 , and print out the values of x 0 ,
x 1 , and x 2 , together with the true solution on the interval [−1, 0]. Verify that the true
2
3
t
t
2
3
solution is x(t) = e + 6 + 6t + 4t + t and y(t) = e − t + t + 2t + 2.