MA2051 - Ordinary Differential Equations
Sample Exam #3 - Solutions
Second Exam - Orginally Given 1996 A Term
- Since N/m, the governing
equation is ,
y(0) = 0.1, y'(0) = 0.
- The mass oscillates because 0.2 y'' + 80 y = 0 has a
characteristic equation with pure imaginary roots: . The general
solution includes and . The period of the
oscillations is or s.
If m = 0.8, then . The
natural frequency is reduced to 10 rad/s; the period increases
- Since a steady state is constant, , and
. This steady state is reasonable
because a) it is negative -- the spring sags when the weight
is added -- and because b) represents a
balance between the spring force due to the extension
and the force of gravity -mg.
- The mass oscillates so long as the characteristic equation of
my'' + p y' + k y = 0 has complex roots; i.e., so long as . Hence, oscillations are possible for N-m/s.
The 0.8 kg mass would oscillate over the same range of p
values as m = 0.2 kg because still holds
with m = 0.8 kg; or you could argue that m = 0.8 kg
oscillates over a larger range of p values, .
- Characteristic equations yields r = -1, -4. Substituting
in the first equation yields .
Substituting in the first equation yields . A general solution is
- Use , , :
- Characteristic equations: . Hence, .
- Need a particular solution. Use undetermined coefficients:
. Find forces A = 4 and . Hence, .
- Use the initial conditions in the previous general solution
to find , . Hence, , , and the solution of the
initial-value problem is .
© 1996 by Will Brother.
All rights Reserved. File last modified on December 6, 1996.