Solved for harmonic oscillator motion using potential method from last
time. Performed time duration integral; learned to make sense of the
sign that arises in the integrand 
, due to the
square root. We noted the time always increases during the motion,
forcing us to make a continuous choice when confronted with periodic
coordinates, as in this problem. Inverting our result for the
integral, we obtained sinusoidal motion with frequency 
,
which we will obtain soon by different methods.
Then began attacking the problem of solving Newton's second law *directly* as a differential equation.
Entered 2 mathematical interludes:
1) Introduced complex numbers, z = x + iy with real components x and
imaginary components y. Rewrote in "phasor" form, 
.
Recalling that 
, this form
has components 
. While
observables are always real, it is often useful to extend the
differential equations we obtain into the complex plane, solve them
there, then apply physical initial conditions to obtain (real)
physical solutions.
2) Differential Equations. Defined differential equations, quickly
specialized to the linear case. First considered the homogeneous case,
which can be written L x(t) = 0, where L is a differential operator
acting on the function x(t). Showed that solutions obey the principle
of superposition: if 
and 
are solutions, then their sum
is also a solution.
-- KB