TEXAS A&M UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 308-501/502
Exam 2 version A, 26 Oct 2012
On my honor, as an Aggie, I have neither given nor received unauthorized aid on this work.
Name (print):
In all questions no analytical work =⇒ no points!
1.
Solve the quadratic equation for x,
x2 − 4x + 3 − 2α − α2 = 0.
Simplify your answer as much as possible.
2.
1. Use Euler’s Formula to show that eiφ = 1 for any real φ (by definition, the absolute
p
value of x + iy is x2 + y 2 ).
π
2. Find the real and imaginary parts of eiφ+i 2 ; simplify as far as possible.
3.
Find the general solution to
1.
y 00 + 3y 0 + 2y = sin(2t),
2.
y 00 + 3y 0 + 2y = e−2t .
4.
Find the general solution of
y 00 − 2y 0 + y =
et
.
1 + t2
5.
We will model the movement of a park swing by the linear equation
mθ00 + γθ0 + kθ = F (t),
m, γ, k > 0,
(1)
where θ(t) is the declination from the vertical. Suppose your movements on the swing
p
create the periodic force F (t) = F0 cos(ω0 t) with the frequency ω0 = k/m.
p
(a) Find the steady state solution. (Hint: because ω0 = k/m things simplify)
(b) What is the phase shift of the solution with respect to the forcing? What is the position and direction of movement of the swing at the time when the force is maximal?
Draw yourself on the swing to illustrate your answer.
Points:
/25