# MA2051 - Ordinary Differential EquationsSample Exam 2 Solutions - A96

### Second Exam - Originally Given 1993 A Term

1
A

dynes/cm; the model is 200 x'' + 24,500 x = 0, x(0) = 0, x'(0) = -6.

B
Use characteristic equations: ; . Hence, a general solution is . The initial conditions force , C2 = -.06/11.07 = -.00542 . Hence, .
C
To find the period, solve to find s.
D
The maximum displacement occurs in when the sine function is . Hence, the maximum displacement (or amplitude of the motion) is 1.85 cm.

2
A
Use undetermined coefficients. First guess . Because these functions are solutions of the homogeneous equation, the correct guess is . Substitute and collect coefficients of sine and cosine to find A = 0, B = 1/4. Hence, .
B
Using the particular solution just found and the given homogeneous solutions, .
C
Because of the leading factor of t in , the solution grows without bound. The system is being forced at its resonant frequency.

3
A
r = -4, 2
B
.
C
Choose , .

4
A
Write .

B
C
Let . Then and

• . Using partial fractions, we find

Then , , and

Hence, the solution is

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