Differential Equations/Existence

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Contents 1 Existence and uniqueness 1.1 Example 9 1.2 Example 10 1.3 Example 11

[edit] Existence and uniqueness

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 y'=f(x,y) 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 y = f(x,y),y(a) = b, 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 (x,y₁) and (x,y₂), then | f(x,y₁) − f(x − y₂ | < K | y₂ − y₁ | for some constant K,

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

So if the Lipschitz condition is satisfied, and, and f(x,y) 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. This solution is usually a trivial solution y(x) = k 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.

[edit] Example 9

y' = ky,y(10) = 500

Is the equation f(x,y) = ky continuous? Yes.

Is the equation continuous? Yes.

So the solution exists and is unique.

[edit] Example 10

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.

[edit] Example 11

Is the equation continuous? Yes.

Is the equation continuous? No. It is discontinuous at y=1

So the solution exists and is not unique. The other solution happens to be the trivial solution y(x) = 1.