This is a differential equation with constant coefficients, with solutions of the form 
. Plugging this form into the ODE, 
. The ODE becomes an algebraic equation for 
:
giving the single linearly independent solution 
. (There is only one solution because the ODE is first order; that is, it has only 1 derivative.) The most general solution is thus
Initial conditions 
imply that 
. Thus 
in agreement with problem set 1, 1a.
This is a differential equation with constant coefficients, with solutions of the form . Plugging this form into the ODE, 
and 
. The ODE becomes an algebraic equation for :
with two solutions 
. We thus have two
linearly independent solutions (since we have a second order ODE), constant and . The most general solution is thus
with derivative
Initial condition 
implies that 
.
Initial condition implies that 
, so 
. Thus
with derivative in agreement with part a.
via
the substitution 
. We thus plug
into our initial differential equation, giving
When 
, we obtain
This is easily solved, giving the cases
imaginary, Underdamped
, Critically Damped
real, Overdamped
discussed in the text.
Thus Newton II gives
Since 
is small, we have the approximation 
, giving the linear ODE
Comparing this to the standard ODE for a damped harmonic oscillator,
gives 
, so that 
and the motion is critically damped. Our general solution is thus
with derivative
At t=0 this gives
Equating to our initial conditions 
gives 
. Thus we have
or a damped oscillator with 
, so that 
, imaginary so that the motion is underdamped. Our linearly independent solutions are thus
which we choose to regroup into linearly independent solutions
We choose this regrouping as 
and 
have quickly understood behavior at t=0, where we will be applying initial conditions. Our
general solution is thus
with derivative
At t=0 this gives
Equating to our initial conditions gives 
, 
. Thus we have
\
