Continued discussion of linear homogeneous differential equations.
Defined order of a D.E. as the highest derivative operator
appearing. An order r D.E. has r different (we used the
fancy term linearly independent) solutions 
. The most general
solution is then a sum of all these different solutions with
undetermined coefficients, 
, where we sum
over all i from 1 to r. It thus takes r different initial
conditions to uniquely determine the motion, by setting the
coefficients 
so that the initial conditions are all true.
We then considered inhomogeneous problems, L x(t) = f(t). Once we find
one particular solution 
such that 
, the sum
solves the problem, where i
ranges from 1 to r and are the homogeneous solutions
above. Again, it takes r initial conditions to determine the
coefficients .
We then specialized further to consider linear differential equations
with constant coefficients. Here solutions x(t) to the homogeneous
problem always take the form 
. We find 
by directly substituting the ansatz into the
differential equation. For a second order homogeneous equation, 
, this gives us the
quadratic equation 
. Depending on the coefficients, we can get a) real,
solutions exponentially grow or decay. If the only homogeneous
solutions present decay, the system approaches its steady state
solution, , after initial "transients" have decayed. b)
imaginary, solutions sinusoidal. c) complex,
we'll see this next lecture.
With all that mathematical preparation, we returned to physics. We
first solved our old friend the simple harmonic oscillator, showing
that NII gave the D.E. 
, for 
defined to equal k/m. We obtained solutions 
, describing sinusoidal motion with frequency
and period 
. We then did some slight
generalizations, allowing the oscillator to have a nonzero equilibrium
point l and to be acted on by a constant force mg. Both of these gave
us inhomogeneous versions of the above D.E., which we solved to find
sinusoidal motion around altered equilibrium points.
-- KB