Math 2250 Lab 10
Name/Unid:
Due Date: 20 November 2014
1. (25 points) This problem is about zero eigenvalues. Let:
y (4) + 2y (3) + y (2) = 6x
(a) Find the homogeneous solution yH (x).
(b) Find a particular solution yP (x). Hint: Try yP (x) = Ax2 + Bx3 .
(c) Find the general solution y(x) = yH (x) + yP (x).
Page 2
2. (25 points) Consider the following series RLC circuit:
R
V
L
C
Suppose a periodic voltage source v(t) = V0 cos(ωt) is applied to the circuit. Using
Kirchoff’s laws, the following differential equation gives the charge q(t) of the capacitor.
L · q 00 (t) + R · q 0 (t) +
1
· q(t) = V0 cos(ωt)
C
Take L = 1 V · s · A−1 , R = 0 Ω, C −1 = 0.36 V · C−1 , and V0 = 3 V. Note: the left sides
of these equations involve variables, while the right sides involve units.
Assume the following initial conditions for q(t):
q(0) = 0
q 0 (0) = 0
(a) What is the natural angular frequency ω0 of this system? Hint: ω0 is the angular
frequency of the solutions to the unforced equation
L · q 00 (t) + R · q 0 (t) +
Page 3
1
· q(t) = 0
C
(b) Assume that ω 6= ω0 . Use the method of undetermined coefficients to solve for the
particular solution qP (t). Then use q(t) = qP (t) + qH (t) to solve the initial value
problem for q(t), where qH (t) is a solution to the homogeneous (unforced) equation.
You may wish to use a computer to check that this solution is correct.
Page 4
(c) Write down the specific case of the solution q(t) (found in part b) for ω = 0.5.
Compute the period of this solution, which is a superposition of two cosine functions.
Use Matlab, Maple, or other technology to graph one period of the solution. What
phenomenon is somewhat exhibited by this solution?
Page 5
(d) Now let ω = ω0 . Use the method of undetermined coefficients to solve for a new
particular solution qP (t). Then use q(t) = qP (t) + qH (t) to solve the initial value
problem for q(t). Graph this solution over the interval 0 ≤ t ≤ 60 seconds. What
phenomenon does this solution exhibit?
Page 6
3. (25 points) Consider the same RLC circuit as in problem 1. For this problem, take
L = 1 V · s · A−1 , R = 1.2 Ω, C −1 = 0.36 V · C−1 , V0 = 3 V, and ω = 1. This gives us
the differential equation:
q 00 (t) + 1.2q 0 (t) + 0.36q(t) = 3 cos(t)
(a) Use the method of undetermined coefficients to find a particular solution qP (t) to
this differential equation.
Page 7
(b) Use the particular solution qP (t) found in part (a) and the solution qH (t) to the
corresponding homogeneous equation to write down the general solution to this
differential equation. Identify the “steady periodic” and “transient” parts of this
general solution.
Page 8
(c) Given the initial conditions
q(0) = 0
q 0 (0) = 0
solve the resulting initial value problem for q(t) satisfying the differential equation
at the beginning of this problem (you may use technology to do this).
Page 9
(d) Graph, on a single plot, the IVP solution found in part (c) as well as the steady
periodic solution identified in part (b). Choose a time interval so that you can
clearly see the convergence of the IVP solution to the steady periodic solution.
Page 10
4. (25 points) The Laplace transform is a function L which takes as input a function of
time f (t) and outputs a function F (s) by the rule:
Z ∞
f (t)e−st dt
L(f )(s) = F (s) =
0
Transform methods are very useful for solving differential equations, but a physical
relationship between F (s) and f (t) is not immediately clear. The goal of this problem
is to give intuition for the variable s.
(a) Consider the following functions:
f1 (t) =
−1 : 0 ≤ t ≤ 1
+1 :
1<t
f2 (t) = cos(2t)
f3 (t) = sin(3t)
Find the Laplace transforms F1 (s), F2 (s), F3 (s) of these functions. It should be
straightforward to compute F1 (s) by hand, but you should look up F2 (s) and F3 (s)
in a table or use a computer.
Page 11
(b) For each of the Laplace transforms found in part (a), replace the variable s by iω.
Here i is the imaginary number i2 = −1, and ω is a real number that represents a
frequency.
Page 12
(c) If z = x + iy is a complex number,
its complex conjugate is z ∗ = x − iy. The non√
negative real number |z| = zz ∗ is called the magnitude of z. Find the magnitude
of F1 (iω), F2 (iω), and F3 (iω). Hint: For this problem, the complex conjugate of
F (iω) is F (−iω), and eix = cos(x) + i sin(x).
Page 13
(d) Plot the magnitudes |F1 (iω)|, |F2 (iω)|, and |F3 (iω)|. Can you relate these plots to
the frequency of f2 (t) and f3 (t)? Based on that relationship, what can you say
about the frequencies of f1 (t)?
Page 14