The differential equation
has characteristic equation
with roots
.
• If
, the roots are real and distinct, and the general solution is
• If
, there is a repeated root and the general solution is
• If
, the roots are a complex conjugate pair
, and the general solution is
which, by Euler's identity, can be expressed as
The solution curve in plot (A) has a somewhat periodic nature to it, so
. The plot suggests that
will oscillate between -∞ and ∞ as
, which tells us
(otherwise, if
the curve would be a simple bounded sine wave, or if
the curve would still oscillate but converge to 0). Since
is the real part of the characteristic root, and we assume
, we have
Since
, we have
The solution curve in plot (B) is not periodic, so
. For
near 0, the exponential terms behave like constants (i.e.
). This means that
• if
, for some small neighborhood around
, the curve is approximately constant,
• if
, for some small neighborhood around
, the curve is approximately linear,
Since
, it follows that
As
, we see
which means the characteristic root is positive (otherwise we would have
), and in turn