Section 2.7 is entitled “Existence and Uniqueness of
Solutions.”
When dealing with initial value problems, we could ask
ourselves two questions:
1. When can we be sure a solution exists at all?
2. How many different solutions are there?
If we were to have these questions, we would be questioning
existence and uniqueness (see what I did there?).
Let’s answer these questions in order by starting with the
issue of existence.
Sometimes initial value problems don’t make sense at all.
You’ll be solving your problem without a care in the world and the problem will
take a left turn into Worry And Devastation and will ask you to divide by zero
or put a negative number into a natural log. In any event, there would be no
solution to your initial value problem and that’s sad.
The book calls these types of initial value problems “anomalies,”
especially when they are in normal form (reminder: x’ = f(t, x) is normal
form).
Nevertheless, here’s a theorem on the existence of an
initial value problem:
Suppose a function f(t, x) is defined and continuous on a
rectangle R in the tx-plane. This means at any point (t0, x0)
that is within that rectangle, the initial value problem x’ = f(t,
x) and x(t0) = x0 has a solution defined that contains
that t0. This solution will be defined until the curve leaves that
rectangle. Note that it is required for the equation to be written in normal
form for this theorem to stand.
Back in section 2.1, we defined the interval of existence as
“the largest interval in which the solution can be defined” (78). Note that the
interval of existence cannot usually be determined from the existence theorem.
Instead, you will have to find the interval by finding an explicit solution.
Let’s focus specifically on linear equations. Recall we
defined linear equations to have the form x’ = a(t)x + g(t). This
would make our normal form f(t, x) for linear equations a(t)x + g(t). If a(t)
and g(t) are continuous for a certain interval of t, this means f is continuous
on a rectangle that as that same interval of t and -∞ < x < ∞. This means
the existence theorem can guarantee the general solution exists over that
entire interval of t. Yay linear equations!
Our existence theorem said that f(t, x), our right hand side
of the equation, must be continuous. Sometimes that isn’t the case and we have
a right hand side that isn’t continuous. However, we still want to solve an
initial value problem. Although these cases do not satisfy our theorem, we want
to break the rules and be rebels and get ourselves an answer. In most of these
cases, the equation is linear [meaning x’ = a(t)x + f(t)] and the
only discontinuity is in f(t). We can agree that the solution x(t) that we find
will be continuous and satisfy everywhere except where f(t) is discontinuous.
Moving onto our second question, which asked how many
solutions are there. In this case, we’re talking about uniqueness. Let us start
with new vocabulary:
If there’s only one solution to a system, then the system
acts the exact same way if it’s started from the same initial conditions. This
system is then called deterministic. If there are multiple solutions, then the
system is unpredictable. Thus for an initial value problem to have a unique
solution, it must also be deterministic.
In the book, there is quite a bit of exposition leading into
the theorem. I’m not including the exposition because it’s basically proving
the theorem and it includes hypotheses and conclusions and stuff. Because this
is technically a summary, we don’t need silly hypotheses and exposition.
Anyway, onto the theorem:
“Suppose f(t, x) and its partial derivative δf/δx are both
continuous on the rectangle R in the tx-plane. Suppose (t0, x0)
[is located on that rectangle R] and that the solutions x’ = f(t, x)
and y’ = f(t, y) satisfy x(t0) = y(t0) = x0.
Then as long as (t, x(t)) and (t, y(t)) stay in R, we have x(t) = y(t)” (82).
The short version of all that says two solutions to the same
equation that start in the same place will travel together. In other words, any
particular point that is contained on R will only have one solution curve. Note
that any point on R can be the starting point.
Obviously, we will want to apply these theorems to initial
value problems. It will be necessary, however, to find a rectangle R in which
the equation satisfies the hypothesis.
The uniqueness theorem also has a geometric interpretation. We’ll
be thinking in terms of the solution curves for this interpretation. If we have
two functions that satisfy the same point, then those functions meet at that
point. Additionally, if these functions are solutions to the same differential
equation, then the uniqueness theorem says that the functions are equal for all
points. This would mean that the graphs would coincide, but solution curves
cannot meet. This means that they cannot cross or touch each other.
Solution curves can get exponentially close to one another,
but the uniqueness theorem states they never meet. Don’t let computer drawn
pictures fool you, then. If you zoom in far enough, you will see that these
curves do not meet.
Extra sources: http://www.math.uiuc.edu/~tyson/existence.pdf
Whew, better you than me....
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