1550078481-Ordinary_Differential_Equations__Roberts_

286 Ordinary Differential Equations

The roots of the auxiliary equation, mr^2 + k = 0, are r 1 = ../klffii = wi

and r 2 = -../klffii = -wi. So the general solution of the differential

equation in (23) is either

where A 1 and A 2 are constants which are to be chosen to satisfy the

initial conditions or equivalently

y(t) = Asin(wt + ¢)

where A and¢ are constants which are to be chosen to satisfy the initial

conditions.

Differentiating the second form of the general solution of the differ-

ential equation, we see that

y'(t) = wAcos (wt+¢).

In order to satisfy the initial conditions given in (23), the constants A

and ¢ must be chosen to simultaneously satisfy

y(O) =A sin¢= -.6 rads and y'(O) = wAcos¢ = .3 rads/s.

Dividing the first equation by the second, we find ¢ must satisfy

sin¢ -.6

---= - = -2 s.

wcos¢ .3

Multiplying by w = ..Jk7ffi = )2.178/.2 = 3.3 rads/s, we see tan¢=

-6.6. So ¢ = -1.42 radians. Next, solving the equation A sin¢ = -.6

for A, yields

A= -.6 = -.6

sin¢ sin( -1.42)

-.6

__

98865

= .607 rads.

Thus, the equation of motion for this pendulum is

y(t) =A sin (wt+¢) = .607 sin (3.3t - 1.42).

b. The amplitude is A = .607 radians.

The period is P = 27r/w = 27r/3.3 = 1.904 seconds.

The frequency is F = 1/ P = .5252 cycles/second.

c. The phase angle is ¢ = -1.42 radians.