Math. 401, Sec. 501 Spring, 2009
Homework 1, due January 28
In Exercises 1 and 2, find approximate solutions of the form x≈x0+ǫx1
(ǫsmall).
1. x3+ǫx2+ 1 = 0
2. x5+ǫx −32 = 0
3. Consider x2+ 2ǫx −1 = 0.
(a) Find approximate solutions of the forms x≈x0+ǫx1and x≈
x0+ǫx1+ǫ2x2.
(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 ǫ= 10, 1, 0.1, and 0.01.
4. Find the second-order solutions (x≈x0+ǫx1+ǫ2x2) to x4+ǫx−1 = 0.