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Suppose I come up with a differential equation: dy/dx = f (x,y)
which is a perfect model for a system I'm trying to study
and I have an initial-value, I know an initial state
and I want to try to make a prediction
by solving an initial-value problem.
Now it would be pretty silly for me to study this problem
if first of all, I didn't know that any solutions existed.
I would spend a long time looking for solutions that don't exist.
Second of all, if I didn't know that a solution to this initial-value problem was unique
then it would be pretty silly for me to try to 'predict' something
because my prediction would be meaningless:
it would depend on which one of the many solutions
I had chosen to make the prediction.
So 'existence' and 'uniqueness' of solutions is an important thing to study
in order to
both solve problems and make predictions.
The main question is this:
if I have an initial-value problem like this.
" When does a solution to the initial-value problem exist? "
and " If it exists, is it a unique solution? "
Fortunately we have easily testable,
sufficient conditions that will tell us 'when' a solution exists
and 'when' it is unique.
First of all
" if f is continuous 'near' the point (a,b) "
" then a solution exists. "
Remember f here is the right-hand side,
defining the ordinary differential equation
'f' is a function of two variables 'x' and 'y'
and we require that it be continuous near the point '(a,b)'
which is the point I'm trying to get the solution
to pass through in order to satisfy the initial-value.
What does this look like?
I have a point in the x-y plane: that's the point (a,b).
So here it is, right there, this little 'orange' dot.
Near (a,b) is some sort of 'area', some blob, some 'neighborhood' around that point,
and I need the function f to be continuous inside that blob.
Now if that's satisfied then I know that, even for just a little..
..maybe just a for a tiny little section,
I can draw my solution through that point (a,b).
Now, what about 'uniqueness'? Who says I can't draw
many 'little solutions' through that point (a,b)?
Well, I need another hypothesis to guarantee uniqueness:
if (also) the partial derivative of f with respect to y
is continuous near (a,b) then the solution will be unique.
So I have to check
not only ..that f is continuous..but also
(if I want to have uniqueness)
that the partial derivative of f with respect y is continuous.
I'd like to point out two subtle aspects of this discussion so far:
The first is: the guarantee of 'existence' and 'uniqueness' is only 'near' (a,b).
In other words
I know the solution exists
but I don't know how 'big' the solution [will become]:
I only know that it exists for a little 'while'
just around at the point (a,b).
Also the uniqueness is only guaranteed near (a,b):
there's nothing to say that the solution
won't be unique for a while and then will split
into multiple pieces somewhere else.
The second subtle thing is that this is not an 'if and only if' [] theorem.
These conditions are not necessary for existence and uniqueness.
So there's nothing to say that even if f is not continuous and even if
'partial f / partial y' is not continuous
I still may have existence and uniqueness.