Exercise 1

Write out the first few terms of the Picard iteration for each of the following

initial value problems. Where possible, find explicit solutions and describe the

domain of this solution.

(a) x

0 = x+ 2; x(0) = 2

(b) x

0 = x

4/3; x(0) = 0

(c) x

0 = x

4/3; x(0) = 1

(d) x

0 = cos(x); x(0) = 0

(e) x

0 = 12x ; x(1) = 1

Exercise 2

Consider the linear system

X

0 = F (X) =

✓0 1�1 0

◆X, X(0) = (1, 0)

Find an analytical solution to the problem, and the first four terms in a Picard

iteration.

Exercise 3

Derive the Taylor series for sin(2t) by applying the Picard method to the first

order system corresponding to the second order initial value problem

x

00 = �4x; x(0) = 0, x

0(0) = 2

Exercise 4

For each of the following functions, find a Lipschitz constant on the region

indicated, or prove that there is none

(a) f(x) = |x| , �1 < x < 1

(b) f(x) = x

1/3, �1 x 1

1