1550078481-Ordinary_Differential_Equations__Roberts_

4

2

y(x) O

• 2

-4

The Initial Value Problem y' = f(x, y); y(c) = d

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-4 - 2 0

x

2 4

Figure 2.6 Direction Field for the DE y' = y^2 and the

Solution of the IVP y' = y^2 ; y(O) = 2

45

You might well ask: "What value is the fundamental existence theorem?
It does not provide a technique for finding the solution to an initial value
problem. Nor does it provide a means of determining the interval on which
the solution exists." The answer is quite simple. The fundamental existence
theorem assures us that there is a solution. It is a waste of time, energy, and
often money to search for a solution when there is none.

In chapter 1, we saw that both y 1 (x) = 0 and Y2(x ) = x^3 were solutions of
the initial value problem

(4) y' = 3y2f3; y(O) = 0.

Notice that f(x, y) = 3y^213 is a continuous function of x and y in any finite

rectangle R which contains (0, 0). So the IVP (4) satisfies the hypotheses of
the fundamental existence theorem and as guaranteed by the theorem there
is a solution to the initial value problem. But, in this instance, there is not
a unique solution. Consequently, some additional condition stronger than
continuity of f(x, y) must be required in order to guarantee the uniqueness of
a solution to an initial value problem.

We state the following existence and uniqueness theorem, again without
proof. This theorem is due to the French mathematician Charles Emile Picard
(1856-1941).