Diff eqn IVP
posted by shasha .
A simple pendulum of length is oscillating through a small angle θ in a medium for which
the resistance is proportional to the velocity. Obtain the differential equation of its motion
and discuss the motion.

This is not an initial value problem!
External forces are zero, so the governing equation is homogeneous:
mx"+Bx'+kx=0
x"=d²x/dt².
x'=dx/dt
x=displacement, positive to the right
m=mass
B=resistance proportional to the velocity
k=stiffness (of a spring), which is resistance proportional to the displacement.
For a pendulum of length L and mass m,
making an angle θ with the vertical, three forces act on the mass m.
The vertical force due to gravity, mg.
A horizontal restoring force (towards the equilibrium position) of mgsin(θ), and the tension, which is equal and opposite to the resultant of the two forces.
The restoring force mgsin(θ) is often approximated by
kx=mgLsin(θ)≅mgLθ for small angles (when θ is in radians). 
why stiffness (of a spring) ?

It is an analogy.
The general equation of motion make an analogy with a mass attached to the end of a spring:
mx"+Bx'+kx=0
where k represents the restoring force per unit displacement. 
oww...ok...
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