Math 4556 Autumn 2022 Problem Set 4
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Math 4556
PRoBLEM SET 4
Autumn 2022
1. #2.5.3
2. #2.5.4
3. (#3.5.3)A little more practice with normal forms.
Show that
= − sin φ + γ sin φ cos φ
dτ
can be written
= Aφ − Bφ3 + O(φ5 )
dτ
near φ = 0. Find A and B . (Note that O(φ5 ) is a sum of higher order terms.)
4. In this problem, you will apply Picard’s Method of Successive Approximation (see next page) to estimate a solution for x˙ = 1 − 2x, where x(0) = 0.
a) Solve the initial value problem.
b) Compute x4 (t) using Picard’s Method of Successive Approximation.
c) Show that your result in part b) approximates your result in part a). Hint: Compute a Taylor series expansion.
5. #3.5.5
6. #3.5.7
Numbers of form #n.n.n denote exercises in the text. These numbers are the same in both editions.
Remarks on Picard’s Method of Successive Approximation
Recall that one approach to proving the Existence and Uniqueness Theorem (for solutions of ordinary differential equations) is to use Picard’s Method of Successive Approximations. This method is based upon the fact that (here, we will consider a one-dimensional phase space, though the theorem and the method are also valid in n dimensions) x(t) is a solution of the initial value problem
x˙ =f(x)
x(0) =x0
if and only if x(t) is a continuous function that satisfies the integral equation
t
x(t) = x0 + f(x(s))ds.
0
The successive approximations to the solution of this integral equation are defined by the sequence of functions (for k = 0, 1, ...)
x0 (t) = x0
t
x1 (t) = x0 + f(x0 (s))ds
0
...
t
xk+1(t) = x0 + f(xk (s))ds
0
The proof of the Existence and Uniqueness Theorem involves proving that the sequence {xk } converges to a unique solution of the differential equation.
One of the benefits of this approach to the proof is that it also makes clear that solutions to the initial value problem depend continuously both upon initial conditions and upon parameter values; we will make frequent use of these facts during the course.
2023-02-07