2.3 2nd order homogeneous linear
differential equations with constant coefficients
II
Now we attack the case of arbitrary
and
in our differential equation
Remember that for
and
this corresponds to the differential equation Eq.
(
2.1
) of the damped linear harmonic
oscillator. We already know that this system performs
oscillations (
) that can be exponentially damped (
). Therefore, we expect something related to
functions. But these are all related to each other if we
recall what we have learned about complex numbers:
Furthermore, for arbitrary complex
,
The function
with complex
comprises the real exponential as well as
and
.
Let us therefore try an
exponential Ansatz in Eq. (
2.24
),
We recognize that
fulfills the differential equation, if the
bracket
is zero:
This is a quadratic equation which in general
has two solutions,
Various cases arrise for different
signs of the argument of the square root (the discriminant).
Let us look at each of these cases in turn.
2.3.1 Positive discriminant
In the case
,
are both real and the two solutions
fulfilling Eq. (
2.24
) are
The general solution is the linear
combination of the two,
In this case there are no oscillations at
all. The ‘damping term’
is too strong.
2.3.2 negative discriminant
In the case
, the two zeros become complex:
where we define an angular frequency
. Now, the two solutions fulfilling Eq. (
2.24
) are
The general solution is the linear
combination of the two,
We rewrite this as
Now, this seems a bit odd since we have got a
complex solution due to the term
. However, the constant coefficients
and
can be complex anyway (and still
is a solution of the differential equation). If we
are only interested in real functions
, we can re–define new constants
and
such that the general solution becomes
Still
and
could be complex numbers, but we can choose them real if
we only want real functions
.
2.3.3 The Marginal Case
For the marginal case
we have only one solution,
, and thus only one integration constant. This is
clearly insufficient! Help can be found in the expression
for the simplest marginal case,
which has as solution
We therefore substitute
and find
Using the differential equation
, we find that
Here we have used
. Thus
2.3.4 Summary
Solutions of
:
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