Assignment 4 Due: Fri., Mar. 12, 2004
1. Which of the following initial-value problems satisfies the conditions of the theorem guaranteeing a unique solution. Justify your answer in each case.
(a) y´ = 3x2 y - 3x5 - 3x3 + 1, 0 < x < 2, y(0) = 1;
(b) y´ = sin(x) y2 - x cos(x), 1 < x < 2, y(1) = -2;
(c) y´ = -cot(x)y + x, -1 < x < 1, y(-1) = 1.
2. Use Euler's method to find approximate solutions to the initial-value problems given in
question 1. Take h = 0.5. What happens when you try to find the solution to the differential equation in part (c)?
3. Use the Taylor method of order 2 to find approximate solutions to the initial-value problems given in question 1 (a) and (b). Take h = 0.5 as before.
4. The exact solution to the differential equation in question 1(a) is x3 + x + 1. Compare your approximate solutions obtained in questions 2 and 3 with this exact solution. Also repeat the calculations with h = 0.25 and compare with the exact solution.
5. Convert the following second-order initial value problem into a system of first-order differential equations and solve them by Euler's method. Take h = 0.25.
y´´ - 2xy´ + 2y = x2, 0 < x < 2, y(0) = 0, y´(0) = 1
6. Use the Runge-Kutta method known as the Midpoint Method to find approximate solutions to the initial-value problems given in question 1 (a) and (b). Take h = 0.5 and compare your results with the exact solution in the case of the first problem.
7. Show that the Taylor polynomial
f(x, y) + h f´(x, y)/2 + h2 f´´(x, y)/6
can be matched to O(h2) excluding the term
by the expression
a1f(x, y) + a2f(x + c, y + df(x, y))
if we choose a1 = 1/4, a2 = 3/4 and c = d = 2h/3. This is the basis of Heun's method.
8. Use the MAPLE routine dsolve to find a numerical solution to the logistics differential equation
y´ = y(3 - y), 0 < x < 4
Take five different initial conditions y(0) = 1, 2, 3, 4, 5 and calculate the solutions for each of these initial conditions at steps of 0.25. Describe the behaviour of the various solutions.
8. Use the MAPLE routine dsolve to find numerical solutions to the following second-order initial-value problems:
(a) y´´ - y´/x + 2xy = sin(x), 1 < x < 2, y(1) = 2, y'(1) = -1
(b) y´´ + cos(x) y´ + x sin(x) y = cot(x), 1 < x < 3, y(1) = 0.5, y'(1) = 1.2
In both cases calculate the solutions at steps of 0.1.
TERM TEST 2 - Wed. March 17, 2004