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By Giuseppe Tenti and annotated by Douglas Wilhelm Harder.

We will now look at 2nd-order differential equations. A general
2nd-order differential equation is of the form

;

however, this cannot be solved in general. We will begin by looking
at specific cases which have a much simpler form.

# Linear Homogeneous 2nd-order Ordinary Differential Equations with
Constant Coefficients

A linear homogeneous 2nd-order differential equation with constant
coefficients is a differential equation of the form:

Equation 5.1.

where
,
, and
are arbitrary constant complex (though usually real)
numbers.

This ODE is fundamental as a description of oscillation and vibration
in all areas of science including electrical, mechanical, but also biological
systems. To solve this, we will first look at the statements made by
Leonhard Euler in
1739 in a letter to
Johann Bernoulli
(paraphrased):

Look,
is not known, so we cannot do anything
about the term
in the differential equation. Therefore,
the only way to make the left hand side of the equation vanish is
to reduce
to something like
and
to something like
, so that the differential equation
becomes something like

and then we choose
and
so that
.

Euler then noticed that this requires that
and its derivatives must be scalar
multiples of each other and the only function where this is possible
are the eigenfunctions of the differential operator, namely,
, where
and
are arbitrary constants.

Under this assumption, we have that
and
and substituting these
into Equation 5.1 yields

and therefore

and therefore, as the exponential function
is not zero, we must require that

.

This is called the *characteristic equation* of the differential
equation and is a quadratic equation with roots

.

There are three cases on the descriminant which
is either positive, zero, or negative yielding two distinct real roots, a double root, or two
complex conjugate roots.

## Case 1:

There are two roots
and
and each is a linearly independent
solution ( is not constant).
It follows that, by the superposition principle, the general solution is

.

#### Example 1

Find the general solution of the differential equation
.

First, the characteristic equation is
which has two roots
and therefore
and
. These two solutions are linearly independent
as
which is not constant.
Therefore, the general solution is

.

Figure 1 shows a number of solutions to this differential equation using various coefficients of
and
.

Figure 1. Numerous solutions to the differential equation of Example 1.

Looking at these, we note that there appear to be two classes of solutions: there is either zero or
one local extreme points and all solutions tend to zero as . These
three cases are shown in Figure 2.

Figure 2. Three representative examples of solutions to Example 1.

## Case 2:

In this case, the two roots are real and coincident: . This
gives us one particular solution
and therefore there must be a second solution which is linearly independent
from this solution.

The trick for finding the second solution was first proposed by D'Alambert in
1748. He suggested that you have two solutions
and
where
is arbitrarily small; therefore,
. Consider first the
two solutions with and
and only once you have the two solutions, let
.

Now, the two linearly independent solutions are

and
.

But any linear combination of these two solutions must also be a solution and in particular,
let us define

.

This must be a solution for any any finite value of
but in the limit, the solution is in an indeterminate form
and therefore we must use l'Hôpital's Rule:

.

The simplest form of the two linearly independent solutions is therefore

.

#### Example 2

Find the general solution of the differential equation
.

First, the characteristic equation is
which has a double root at
.
The two linearly independent solutions are, therefore,
and ; hence, the general solution is

.

Figure 3 shows a number of solutions to this differential equation using various coefficients of
and
.

Figure 3. Numerous solutions to the differential equation of Example 2.

As with the previous case, there are either zero or
one local extreme points and all solutions tend to zero as . These
are the same three cases already shown in Figure 2.

## Case 3:

In this case, the roots of the characteristic equation are complex conjugates of the form
where
and
. The two linearly independent
solutions using these functions are

and ;

however, these are complex-valued functions, yet any linear combination of these solutions is also
a solution. This freedom allows us to find two **real-valued** linearly independent solutions:

- Sum the solutions:
- Take the difference:

The constants
and
may be ignored and hence we may conclude that two real-valued linearly independent solutions are
and
. The superposition principle gives us that the general solution is

.

As an alternate formulation, we may write this using two constants in the amplitude-phase form:

where is the amplitude and
is the phase shift. This can also, of course, be written in terms of a sine
function together with the corresponding adjusted phase shift.

#### Example 3

Find the general solution of the differential equation
.

First, the characteristic equation is
which has two complex roots at .
The two linearly independent solutions are, therefore,
and ;
hence, the general solution is

The two linearly independent solutions are, therefore,

and

; hence, the general solution is

,

or, an alternate formulation is .

Figure 4 shows a number of solutions to this differential equation using various coefficients of
and
while Figure 5 shows the two linearly independent solutions. All solutions oscillate infinitely many times.

Figure 4. Numerous solutions to the differential equation of Example 3.

Figure 5. The two linearly independent solutions to the differential equation of Example 3.

## Examples with Animations

The reader may wonder about the transition from real to complex roots
of this system, and therefore, the three animations shown in Figures 6-8 will show
how the solutions to a differential equation change as the parameters
change. In all cases, the differential equation will start with two
different real roots, will make the transition to two identical real
roots with the differential equation

and then continue with two complex roots. In each case, the reader
will note that there is a smooth transition from a solution with no extreme
points past to a solution which is oscillating.

Figure 6. Solutions to the differential equation

for values of
.

Figure 7. Solutions to the differential equation

for values of
.

Figure 8. Solutions to the differential equation
for values of .

Another possibility is the transition from a solution with no roots
past , to a solution with one real root
past zero, to a solution which is oscillating. There do not exist
solutions where there are two extreme points:

- The solution will decay towards the zero solution,
- The solution may pass the zero solution once and then decay towards it, or
- The solution will oscillate infinitely many times around the
zero solution.

This natural and smooth transition from one type of solution to another should
be expected, for example, calculating the integral
is a polynomial for almost all values
of except when of in which
case the integral is a logarithm, and yet, there is a smooth transition from
one state to the other, as is shown in Figure 9.

Figure 9. The integral for values of
.