# MA2051 - Ordinary Differential EquationsSample Exam #3 - Solutions

Second Exam - Orginally Given 1996 A Term

1. Since N/m, the governing equation is , y(0) = 0.1, y'(0) = 0.
2. 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 to s.

3. 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.
4. 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, .

1. Characteristic equations yields r = -1, -4. Substituting in the first equation yields . Substituting in the first equation yields . A general solution is

2. Use , , :

1. Characteristic equations: . Hence, .
2. Need a particular solution. Use undetermined coefficients: . Find forces A = 4 and . Hence, .
3. Use the initial conditions in the previous general solution to find , . Hence, , , and the solution of the initial-value problem is .

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