|Dec24-12, 08:51 PM||#1|
Can someone give me a qualitative example of the uniqueness theorem of a first order linear differential equation? I have read the definition, but I am not 100% positive of what it means in regards to an initial value problem.
Im confused about what a unique solution is when/if you change the initial value. Would it not have many different solutions?
|Dec25-12, 07:20 AM||#2|
Yes, of course if you change the initial conditions, you will have different solutions! What the "existance and uniqueness" theorem for initial value problems says is that a given (well behaved) differential equation, with specific initial conditions will have a unique solution.
Specifically, the basic "existance and uniqueness theorem" for first order equations, as given in most introductory texts, says
"If f(x, y) is continuous in x and y and "Lipschitz" in y in some neighborhood of [itex](x_0, y_0)[/itex] then the differential equation dy/dx= f(x,y) with initial value [itex]y(x_0)= y_0[/itex] has a unique solution in some neighborhood of [itex]x_0[/itex]".
(A function, f(x), is said to be "Lipschitz" in x on a neighborhood if there exists some constant C so that |f(x)- f(y)|< C|x- y| for all x and y in that neighborhood. One can show that all functions that are differentiable in a given neighborhood are Lipschitz there so many introductory texts use "differentiable" as a sufficient but not necessary condition.)
We can extend that to higher order equations, for example [itex]d^2y/dx^2= f(x, y, dy/dx)[/itex] by letting u= dy/dx and writing the single equation as two first order equations, dy/dx= u and du/dx= f(x, y, u). We can then represent those equations as a single first order vector equation by taking [itex]V= <y, u>[/itex] so that [itex]dV/dx= <dy/dx, du/dx>= <u, f(x,y,u)>[/itex]. Of course, we now need a condition of the form [itex]V(x_0)= <y(x_0), u(x_0)>[itex] is given which means that we must be given values of y and its derivative at the same value of [itex]x_0[/itex], not two different values.
For example, the very simple equation [itex]d^2y/dx^2+ y= 0[/itex] with the boundary values y(0)= 0, [itex]y'(\pi/2)= 0[/itex] does NOT have a unique solution.
Again, the basic "existance and uniqueness theorem" for intial value problems does NOT say that there exist a unique solution to a differential equation that will work for any initial conditions. It says that there exists a unique solution that will match specific given initial conditions.
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