# MA2051 - Ordinary Differential EquationsSample Makeup Exam Solutions - B96

Second Exam - Originally Given 1996 A Term

1
1
, , .

Solve the preceding initial-value problem via characteristic equations: . Hence, ; ; . Since , the maximum angular displacement is 0.05 radians.

2
From the preceding solution formula, the period of the pendulum is or s. The pendulum will first return to vertical in half this time (it passes vertical twice in each period) or in s.

Since period is independent of mass, changing the mass will have no effect.

3
Repeating the analysis of part (b) with L in place of 2.45 m shows that . The period is or . Hence, increasing L increases the period.

4
The characteristic equation of is . The characteristic roots are

Since p is small and both m and L can be made large, ; these roots are complex. Their real part is -p/2mL; solutions of this differential equation will be damped by the exponential factor . To minimize the effects of this term, choose both m and L as large as possible.

2
Write the second-order equation as the system Use , , . Then

3
Characteristic equations: leads to . When r = 8, choose p = 1 and find . When r = -12, choose p = 1 and find . Hence,

4
1
Characteristic equations: : .

2
Undetermined coefficients: . Then forces -96 B = 24 or B = -1/4. Then 4 B - 96 A = 0 yields A = B/24 = -1/96 and . Hence, .

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