Math. 401, Sec. 501 Spring, 2006
Homework 1, due January 25
In Exercises 1 and 2, find approximate solutions of the form xโx0+๎x1
(๎small).
1. x3+๎x2+1=0
2. x5+๎x โ32 = 0
3. Consider x2+2๎x โ1=0.
(a) Find approximate solutions of the forms xโx0+๎x1and xโ
x0+๎x1+๎2x2.
(b) Check the consistency of your answers to (a) with the Taylor ex-
pansion of the exact solution.
(c) Compare the first-order, second-order, and exact solutions numer-
ically, for ๎= 10, 1, 0.1, and 0.01.
4. Find the second-order solutions (xโx0+๎x1+๎2x2)tox
4+๎x โ1=0.