Existence and Uniqueness of Solutions
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.
. Assume f has a partial derivative with respect to y and
that 
is also continuous
on the rectangle R. Then there exists an interval 
(with 
) such that the initial value problem
has a unique solution y(x) defined on the interval I.
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
It is not hard to see in fact that if a function y(x) satisfies the equation (called functional equation)
on an interval I, then it is solution to the initial value problem
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:
is known, define the
function
which, under the assumptions made on f(x,y),
converges to the solution y(x) of the initial value problem
For more on this, check the page Picard Iterative Process.
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Author: Mohamed Amine Khamsi