We begin now our study of the Picard Theorem giving Existence and Uniqueness for the general initial value problem
y' = f(t,y), y(t0) = Y (*)
There are several important aspects to the theorem. Here is a summary:
The proof of the Picard Theorem involves introducing a sequence of functions, the Picard iterates yn, that approximate the solution of (*). They are defined by the scheme yn' = f(t,yn - 1), yn(t0) = Y, and starting off the iteration with the zero function y0(t) = 0.
The theorem asserts that there is a uniques solution for |t - t0| < h. The number h can be computed as h = min(a, b/M) where M is an upper bound on |f(t,y)| on the region of definition. It is important that you can compute M by maximization techniques or using inequalities, and hence you can actually find h. And h can be much smaller than a, the width of the original t interval.
The theorem asserts that the Picard iterates converge uniformly. This is a very strong type of convergence and it gives you extra properties of the limit functions, i.e. if each yn is continuous, so is their limit (and similar features for their derivatives).
The proof of the theorem involves an estimate of the degree to which the Picard iterates approximate the unique solution of (*). This estimate is an error estimate for predicting how closely we can approximate solutions of (*) using Picard iterates.
The proof of uniqueness is quite independent of the rest of the proof. It uses differential inequalities, a subject that is very powerful and of value in many areas of differential equations. You see here only the most elementary steps into that subject, but they deliver uniqueness rather easily.
You are responsible for mastering the material on Euler equations, section 5.5. The idea in an Euler equation is to look for solutions of the form y = (x - x0)r where r is an unknown constant. When you substitute this in an Euler equation (centered at x0) you get a quadratic equation for r. You should learn how to identify an Euler equation and be able to solve such equations when the r's are real and unequal, both the same, or complex conjugates of each others. Do a couple of problems of each type to be sure you are prepared for an Euler equation on the next test!
1. (pg 241 no 12):
Solve the following differential equation by power series methods
about x0, finding the recurrence relation and the
first 4 terms in each of two linearly independent solutions. If
possible, find the general term in each solution: (1 -
x)y'' + xy' - y = 0, x0
= 0.
2. (pg 241 no 16):
For the initial value problem (2 + x2)y''
- xy' + 4y = 0, y(0) = -1, y'(0) = 3
a) Find the first five non-zero terms in a power series solution of
this problem,
b) Plot the four term and five term approximations to the solution on
the same graph (accurately!),
c) From the plot estimate an interval in which the four term
approximation is reasonably accurate.
3. (pg 247 no 5):
Determine a lower bound for the radius of convergence of power
solutions of the following equation centered at the points
x0 given: y'' + 4y' + 6xy = 0,
x0 = 0, x0 = 4.
4. (pg 247 no 8):
Determine a lower bound for the radius of convergence of power
solutions of the following equation centered at the points
x0 given: xy'' + y = 0,
x0 = 1.
5. (pg 106 no 9):
Use the method of Picard Iteration to approximate the solution of the
initial value problem
y' = t2 + y2, y(0) = 0
i.e. a) calculate the Picard iterates y1,
y2, y3, given that
y0 = 0
b) Plot y1(t), y2
(t), y3 (t) and observe whether
the iterates appear to be converging.