Nonhomogeneous
Differential Equations
It’s now time to start thinking about how to solve
nonhomogeneous differential equations. A
second order, linear nonhomogeneous differential equation is
where g(t) is a
nonzero function. Note that we didn’t
go with constant coefficients here because everything that we’re going to do in
this section doesn’t require it. Also,
we’re using a coefficient of 1 on the second derivative just to make some of
the work a little easier to write down.
It is not required to be a 1.
Before talking about how to solve one of these we need to
get some basics out of the way, which is the point of this section.
First, we will call
the associated homogeneous differential equation to (1).
Now, let’s take a look at the following theorem.
Theorem
Suppose that Y_{1}(t)
and Y_{2}(t) are two
solutions to (1)
and that y_{1}(t) and y_{2}(t) are a fundamental set
of solutions to the associated homogeneous differential equation (2)
then,
is a solution to (2)
and it can be written as

Note the notation used here.
Capital letters referred to solutions to (1)
while lower case letters referred to solutions to (2). This is a fairly common convention when
dealing with nonhomogeneous differential equations.
This theorem is easy enough to prove so let’s do that. To prove that Y_{1}(t)  Y_{2}(t)
is a solution to (2)
all we need to do is plug this into the differential equation and check it.
We used the fact that Y_{1}(t)
and Y_{2}(t) are two
solutions to (1)
in the third step. Because they are
solutions to (1)
we know that
So, we were able to prove that the difference of the two
solutions is a solution to (2).
Proving that
is even easier. Since
y_{1}(t) and y_{2}(t) are a fundamental set
of solutions to (2)
we know that they form a general solution and so any solution to (2)
can be written in the form
Well, Y_{1}(t)
 Y_{2}(t) is a solution to (2),
as we’ve shown above, therefore it can be written as
So, what does this theorem do for us? We can use this theorem to write down the
form of the general solution to (1). Let’s suppose that y(t) is the general solution to (1) and
that Y_{P}(t) is any solution
to (1)
that we can get our hands on. Then using
the second part of our theorem we know that
where y_{1}(t)
and y_{2}(t) are a
fundamental set of solutions for (2). Solving for y(t) gives,
We will call
the complementary solution and Y_{P}(t) a particular solution. The general solution to a differential
equation can then be written as.
So, to solve a nonhomogeneous differential equation, we will
need to solve the homogeneous differential equation, (2),
which for constant coefficient differential equations is pretty easy to do, and
we’ll need a solution to (1).
This seems to be a circular argument. In order to write down a solution to (1)
we need a solution. However, this isn’t
the problem that it seems to be. There
are ways to find a solution to (1). They just won’t, in general, be the general
solution. In fact, the next two sections
are devoted to exactly that, finding a particular solution to a nonhomogeneous
differential equation.
There are two common methods for finding particular solutions
: Undetermined Coefficients and Variation of Parameters. Both have their advantages and disadvantages
as you will see in the next couple of sections.