Existence and Uniqueness of Solutions

Existence and uniqueness theorem is the tool which makes it possible for us to conclude that there exists only one solution to a first order differential equation which satisfies a given initial condition. How does it work? Why is it the case? We believe it but it would be interesting to see the main ideas behind. First let us state the theorem itself.

Note that the number h may be smaller than a. In order to understand the main ideas behind this theorem, assume the conclusion is true. Then if y(x) is a solution to the initial value problem, we must have

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It is not hard to see in fact that if a function y(x) satisfies the equation (called functional equation)

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on an interval I, then it is solution to the initial value problem

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Picard was among the first to look at the associated functional equation. The method he developed to find y is known as the method of successive approximations or Picard's iteration method. This is how it goes:

Step 1. Consider the constant function

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Step 2. Once the function tex2html_wrap_inline67 is known, define the function

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Step 3. By induction, we generate a sequence of functions tex2html_wrap_inline71 which, under the assumptions made on f(x,y), converges to the solution y(x) of the initial value problem

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For more on this, check the page Picard Iterative Process.

If you would like to practice more, click on Example for more examples.

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Author: Mohamed Amine Khamsi

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