Before we start off this section we need to make it very clear
that we are only going to scratch the surface of the topic of boundary value
problems. There is enough material in
the topic of boundary value problems that we could devote a whole class to it. The intent of this section is to give a brief
(and we mean very brief) look at the idea of boundary value problems and to
give enough information to allow us to do some basic partial differential
equations in the next chapter.
Now, with that out of the way, the first thing that we need
to do is to define just what we mean by a boundary value problem (BVP for
short). With initial value problems we
had a differential equation and we specified the value of the solution and an
appropriate number of derivatives at the same point (collectively called
initial conditions). For instance for a
second order differential equation the initial conditions are,
With boundary value problems we will have a differential
equation and we will specify the function and/or derivatives at different points, which we’ll call
boundary values. For second order
differential equations, which will be looking at pretty much exclusively here,
any of the following can, and will, be used for boundary conditions.


(2)



(3)

As mentioned above we’ll be looking pretty much exclusively
at second order differential equations.
We will also be restricting ourselves down to linear differential
equations. So, for the purposes of our
discussion here we’ll be looking almost exclusively at differential equations
in the form,
along with one of the sets of boundary conditions given in (1)
(4). We will, on occasion, look at some different
boundary conditions but the differential equation will always be on that can be
written in this form.
As we’ll soon see much of what we know about initial value
problems will not hold here. We can, of
course, solve (5)
provided the coefficients are constant and for a few cases in which they
aren’t. None of that will change. The changes (and perhaps the problems) arise
when we move from initial conditions to boundary conditions.
One of the first changes is a definition that we saw all the
time in the earlier chapters. In the
earlier chapters we said that a differential equation was homogeneous if for all x. Here we will say that a boundary value
problem is homogeneous if in
addition to we also have and (regardless of the boundary conditions
we use). If any of these are not zero we
will call the BVP nonhomogeneous.
It is important to now remember that when we say homogeneous
(or nonhomogeneous) we are saying something not only about the differential
equation itself but also about the boundary conditions as well.
The biggest change that we’re going to see here comes when
we go to solve the boundary value problem.
When solving linear initial value problems a unique solution will be
guaranteed under very mild conditions.
We only looked at this idea for first order IVP’s but the idea does
extend to higher order IVP’s. In that section we saw that all we needed to
guarantee a unique solution was some basic continuity conditions. With boundary value problems we will often
have no solution or infinitely many solutions even for very nice differential
equations that would yield a unique solution if we had initial conditions
instead of boundary conditions.
Before we get into solving some of these let’s next address
the question of why we’re even talking about these in the first place. As we’ll see in the next chapter in the
process of solving some partial differential equations we will run into
boundary value problems that will need to be solved as well. In fact, a large part of the solution process
there will be in dealing with the solution to the BVP. In these cases the boundary conditions will
represent things like the temperature at either end of a bar, or the heat flow
into/out of either end of a bar. Or
maybe they will represent the location of ends of a vibrating string. So, the boundary conditions there will really
be conditions on the boundary of some process.
So, with some of basic stuff out of the way let’s find some
solutions to a few boundary value problems.
Note as well that there really isn’t anything new here yet. We know
how to solve the differential equation and we know how to find the constants by
applying the conditions. The only
difference is that here we’ll be applying boundary conditions instead of
initial conditions.
Example 1 Solve
the following BVP.
Solution
Okay, this is a simple differential equation to solve and
so we’ll leave it to you to verify that the general solution to this is,
Now all that we
need to do is apply the boundary conditions.
The solution is
then,

We mentioned above that some boundary value problems can
have no solutions or infinite solutions we had better do a couple of examples
of those as well here. This next set of
examples will also show just how small of a change to the BVP it takes to move
into these other possibilities.
Example 2 Solve
the following BVP.
Solution
We’re working with the same differential equation as the
first example so we still have,
Upon applying
the boundary conditions we get,
So in this
case, unlike previous example, both boundary conditions tell us that we have
to have and neither one of them tell us anything
about . Remember however that all we’re asking for
is a solution to the differential equation that satisfies the two given
boundary conditions and the following function will do that,
In other words,
regardless of the value of we get a solution and so, in this case we
get infinitely many solutions to the boundary value problem.

Example 3 Solve
the following BVP.
Solution
Again, we have the following general solution,
This time the
boundary conditions give us,
In this case we
have a set of boundary conditions each of which requires a different value of
in order to be satisfied. This, however, is not possible and so in
this case have no solution.

So, with Examples 2 and 3 we can see that only a small
change to the boundary conditions, in relation to each other and to Example 1,
can completely change the nature of the solution. All three of these examples used the same
differential equation and yet a different set of initial conditions yielded, no
solutions, one solution, or infinitely many solutions.
Note that this kind of behavior is not always unpredictable
however. If we use the conditions and the only way we’ll ever get a solution to the
boundary value problem is if we have,
for any value of a. Also, note that if we do have these boundary
conditions we’ll in fact get infinitely many solutions.
All the examples we’ve worked to this point involved the
same differential equation and the same type of boundary conditions so let’s
work a couple more just to make sure that we’ve got some more examples
here. Also, note that with each of these
we could tweak the boundary conditions a little to get any of the possible
solution behaviors to show up (i.e.
zero, one or infinitely many solutions).
Example 4 Solve
the following BVP.
Solution
The general solution for this differential equation is,
Applying the
boundary conditions gives,
In this case we
get a single solution,

Example 5 Solve
the following BVP.
Solution
Here the general solution is,
and we’ll need
the derivative to apply the boundary conditions,
Applying the
boundary conditions gives,
This is not
possible and so in this case have no
solution.

All of the examples worked to this point have been
nonhomogeneous because at least one of the boundary conditions have been
nonzero. Let’s work one nonhomogeneous example
where the differential equation is also nonhomogeneous before we work a couple
of homogeneous examples.
Example 6 Solve
the following BVP.
Solution
The complementary solution for this differential equation
is,
Using Undetermined Coefficients or Variation of Parameters it is easy to
show (we’ll leave the details to you to verify) that a particular solution
is,
The general
solution and its derivative (since we’ll need that for the boundary
conditions) are,
Applying the
boundary conditions gives,
The boundary
conditions then tell us that we must have and they don’t tell us anything about and so it is can be arbitrarily chosen. The solution is then,
and there will
be infinitely many solutions to the BVP.

Let’s now work a couple of homogeneous examples that will
also be helpful to have worked once we get to the next section.
Example 7 Solve
the following BVP.
Solution
Here the general solution is,
Applying the
boundary conditions gives,
So is arbitrary and the solution is,
and in this
case we’ll get infinitely many solutions.

Example 8 Solve
the following BVP.
Solution
The general solution in this case is,
Applying the
boundary conditions gives,
In this case we
found both constants to be zero and so the solution is,

In the previous example the solution was . Notice however, that this will always be a
solution to any homogenous system given by (5) and
any of the (homogeneous) boundary conditions given by (1) (4). Because of this we usually call this solution
the trivial solution. Sometimes, as in the case of the last example
the trivial solution is the only solution however we generally prefer solutions
to be nontrivial. This will be a major
idea in the next section.
Before we leave this section an important point needs to be
made. In each of the examples, with one
exception, the differential equation that we solved was in the form,
The one exception to this still solved this differential
equation except it was not a homogeneous differential equation and so we were
still solving this basic differential equation in some manner.
So, there are probably several natural questions that can
arise at this point. Do all BVP’s
involve this differential equation and if not why did we spend so much time
solving this one to the exclusion of all the other possible differential
equations?
The answers to these questions are fairly simple. First, this differential equation is most
definitely not the only one used in boundary value problems. It does however exhibit all of the behavior
that we wanted to talk about here and has the added bonus of being very easy to
solve. So, by using this differential
equation almost exclusively we can see and discuss the important behavior that
we need to discuss and frees us up from lots of potentially messy solution
details and or messy solutions. We will,
on occasion, look at other differential equations in the rest of this chapter,
but we will still be working almost exclusively with this one.
There is another important reason for looking at this
differential equation. When we get to
the next chapter and take a brief look at solving partial differential
equations we will see that almost every one of the examples that we’ll work
there come down to exactly this differential equation. Also, in those problems we will be working
some “real” problems that are actually solved in places and so are not just
“made up” problems for the purposes of examples. Admittedly they will have some
simplifications in them, but they do come close to realistic problem in some
cases.