Second-order linear equations with constant coefficients are very important,
for applications in mechanical and electrical engineering (as we will see).
The general second-order constant-coefficient linear equation is
, where and are constants. We will
be especially interested in the cases where either (the
homogeneous case) or
for some constant .
The main idea in solving these equations is:
make an educated guess at the form of the solution, and see what has to
happen to make a function of this form satisfy the equation. In the
it turns out that the form to use is
, where is a constant. If we plug this in to the
differential equation, we get
, and factoring out
the (which is not ) we are left with
This is called the characteristic equation of the operator (or of the
differential equation). If is a root of the characteristic equation,
is a solution of the differential equation.
Recall that we are looking for a fundamental set consisting of two
linearly independent solutions. Now a quadratic equation may have two
real roots, and this would give us our two solutions (it is not hard to see
that they are linearly independent).
Example: Consider the equation
has two roots and
. Therefore and form the fundamental
set of solutions. The general solution is
Now, a quadratic equation may also have only one real root, or no real
roots. We will have to deal with these possibilities as well.
At this point it is useful to introduce the differentiation
operator defined by . We can then write the operator
of our differential equation as
, a polynomial in . We can manipulate polynomials
in just as we would manipulate ordinary polynomials. For example,
we might factor
What does mean? These are operators, so they are defined
by what they do to functions. The ``multiplication'' of operators
is really composition: you first let the operator on the right
act on the function, and then the operator on the left acts on the
result. Thus for you first take
The characteristic polynomial is related to the way the operator
acts on exponential functions:
So, for example,
. Or in general, for any polynomial ,
This has the following consequences:
Now how can we deal with the homogeneous equation when
has only one real root, or with the non-homogeneous equation
where ? The key is the
Exponential Shift Theorem: For any polynomial , constant
and function ,
- If then is a solution of the
homogeneous equation .
- If then
one solution of the non-homogeneous equation
Proof: Note that
Repeated application of this gives us the result for any power of , e.g.
Adding constants times powers of gives us all polynomials.
Similarly, we can try
where . Again we try
, and get
, so we want
Now let's apply this to where the polynomial
has only one root.
where is the root. We look for a solution
of the form
for some function . The Exponential Shift
, so we want .
Two linearly independent solutions of this are and .
Thus a fundamental set of solutions of is
Example: Solve the initial value problem
Writing we want . Clearly one solution of this
is the constant . Then from we get a solution
. Clearly one solution
is , i.e.
, with one root .
Since the right side of the equation involves , we use the
Exponential Shift Theorem with
. We have
, so one solution is . This gives
us one solution of our differential equation:
. The fundamental set of
solutions of the homogeneous equation is
and . Thus the general solution is
. From the initial conditions we
. Thus and
, and the answer is