Ordinary Differential Equations/Existence

Existence and uniquenessEdit

So, does this mean that if we have an initial condition we will always have 1 and only 1 solution? Well, not exactly. Its still possible in some circumstances to have either none or infinitely many solutions.

We will restrict our attention to a particular rectangle for the differential equation   where the solution goes through the center of the rectangle. Let the height of the rectangle be h, and the width of the rectangle be w. Now, let M be the upper bound of the absolute value of f(x,y) in the rectangle. Define b to be the smaller of w and h/M to ensure that the function stays within the rectangle.

Existence Theorem: If we have an initial value problem  , we are guaranteed a solution will exist if f(x,y) is bounded on some rectangle I surrounding the point (a,b).

Basically this means that so long as there is no discontinuity at point (a,b), there is at least 1 solution to the problem at that point. There can still be more than 1 solution, though.

Uniqueness Theorem: If the following Lipschitz condition is satisfied as well

For all x in the rectangle, then for two points   and  , then   for some constant  ,

then the solution is unique on some interval   containing x=a.

So if the Lipschitz condition is satisfied, and, and   is bounded, there is a solution and the solution is unique. If the Lipschitz condition is not satisfied, there is at least 1 other solution[citation needed]. This solution is usually a trivial solution   where k is a constant.

We will use two different methods for proving these theorems. The first method is the Method of Successive Approximations and the second method is the Cauchy Lipschitz Method."

Lets try a few examples.


Example 9Edit

 

Is the equation   continuous? Yes.

Is the equation   continuous? Yes.

So the solution exists and is unique.


Example 10Edit

 

Is the equation   continuous? No. There is a discontinuity at x=0. If we used any other point it would exist.

So the solution does not exist.

Example 11Edit

 

Is the expression   continuous? Yes.

Is the expression   continuous for y(1)=1? No. It is discontinuous at y=1 (that is, it is not defined at   and is unbounded as  ), but continuous for all x.

Lifschitz condition is not satisfied, though the existence condition is satisfied. Hence, the solution exists but need not be unique.

The first solution is  .

The other solution happens to be the trivial one,  .