Download Exam 3

APPM 2360: Section exam 3
July 8, 2011.
ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your instructor’s name, and
(3) a grading table. Text books, class notes, and calculators are NOT permitted. A one-page crib
sheet is allowed.
Problem 1: (20 points)
(a) Find the general solution for the following homogeneous linear differential equations. Express
all solutions in terms of real functions.
00
0
(i) 5y − 10y + 5y = 0
00
(ii) y + 4y = 0
(iii) 2y (4) − 8y (3) + 8y (2) = 0
(b) Convert the following simple harmonic oscillator equation,
2ẍ + 4ẋ + x = 12cos(8t),
into a first-order system of differential equations. That is, rewrite it in the following form
x˙1
x1
f1 (t)
=A
+
x˙2
x2
f2 (t)
where A ∈ M2×2 , x1 = x and x2 = ẋ.
Problem 2: (20 points) Consider the second-order, constant-coefficient, forced linear ODE
y 00 − y 0 − 6y = f (t)
(a) Find the homogeneous solution yh (t).
(b) For the following forcing functions f (t), what is the correct form/guess for the particular
solution yp (t) for the equation above? DO NOT SOLVE!!!
(i) f (t) = t2 − 1,
(ii) f (t) = te−2t ,
(iii) f (t) = sin(2t).
Problem 3: (20 points) Consider the following Euler-Cauchy equation
dy
d2 y
− 2t + 2y = f (t), t > 0
2
dt
dt
(a) Let f (t) = 0. The homogeneous solutions take the form yh (t) = tr . Substitute this form into
the ODE and use it to find two such solutions.
(b) Verify that the two homogeneous solutions you found in part (i) are linearly independent then
determine the general solution for this homogeneous problem.
(c) Now let f (t) = t3 cos(t). Find a particular solution yp (t) to this ODE using variation of
parameters. Keep in mind that y1 and y2 are of the form tr (found in part (a)) not ert .
t2
PLEASE TURN OVER
Problem 4: (20 points) Short answers: provide an answer with a brief explanation. (Each part
is doable with little-to-no work!!! )
(a) How many linearly independent solutions are needed to form a basis for the solution space of
0
the third-order
linear homogeneous ODE y 000 − y 00 −
2y = 0?
2
5 6
(b) ~v =
is an eigenvector of the matrix A =
. What is the corresponding eigenvalue?
1
2 4
(c) Suppose λ = −2 is an eigenvalue of the matrix B with corresponding eigenvector ~v . What is
one eigenvalue of the matrix B 2 ? (Hint: ~v is an eigenvector of B 2 too! )
(d) Suppose λ = 0 is an eigenvalue of the matrix C. Is C invertible or not invertible?
Problem 5: (20 points) Consider the following matrix


3 0 0
A = −7 2 3
5 3 2
(a) Find all eigenvalues of A.
(b) What is the eigenspace Eλ=3 ? What is its dimension?