Consider the initial value problem dydx x2 y and y2 2 a U

Consider the initial value problem
dy/dx = x^2 + y and y(-2) = -2.
(a) Use Picard\'s Theorem to show that the initial value problem has a unique solution.
(b) Verify that y(x) = - 2 - 2x - x2 is a solution to the initial value problem.
(c) Let h = 0:1. Use Euler\'s Method to estimate y(8). (Uses 100 steps.)
(d) Graph the points from Euler\'s Method, and the explicit solution together and com-
pare.
(e) Plot the vector eld in Mathematica with the curves and the initial conditions
y(-2) = -2, y(-2) = -2.1 and y(-2) = y(-1:9). What happens as you change
the initial condition even slightly?
(f) How does Euler\'s solution compare to the solution curve produced by Maple?

Solution

Picard Theorem: If a function f : C C is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point.

Sketch of Proof: Picard\'s original proof was based on properties of the modular lambda function, usually denoted by , and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc. This function is explicitly constructed in the theory of elliptic functions. If f omits two values, then the composition of f with the inverse of the modular function maps the plane into the unit disc which implies that f is constant by Liouville\'s theorem.

This theorem is a significant strengthening of Liouville\'s theorem which states that the image of an entire non-constant function must be unbounded. Many different proofs of Picard\'s theorem were later found and Schottky\'s theorem is a quantitative version of it. In the case where the values of f are missing a single point, this point is called a lacunary valueof the function.

Great Picard\'s Theorem: If an analytic function f has an essential singularity at a point w, then on anypunctured neighborhood of w, f(z) takes on all possible complex values, with at most a single exception, infinitely often.

This is a substantial strengthening of the Casorati-Weierstrass theorem, which only guarantees that the range of f is dense in the complex plane. A result of the Great Picard Theorem is that any entire, non-polynomial function attains all possible complex values infinitely often, with at most one exception.

The \"single exception\" is needed in both theorems, as demonstrated here: