Math 33B Exam 2 Solutions
Problem 1.
(A) (2 points) For which are the functions y1 (t) = cos(t) and y2 (t) = sin(t+) not linearly
independent?
3
0, , , , 2
2
2
Solution: /2 and 3/2. For these values o
Math 33B Quiz 7B Solution
Name:
SID:
Problem 1. Compute the general solution, then classify and sketch the phase portrait for
the system
5 2
0
y =
y.
6 2
Solution: The characteristic polynomial is fou
Math 33B Quiz 8B
Name:
SID:
Problem 1. Compute the matrix exponential etA , where
2 1 0
0
1
A= 0
0 4 4
Solution: The eigenvalue of A is 2, with algebraic multiplicity equal to 3. The only
independent
Math 33B Quiz 8A
Name:
SID:
Problem 1. Compute the matrix exponential etA , where
1 1 0
A = 1 0 1
1 2 2
Solution: The eigenvalue of A is 1, with algebraic multiplicity equal to 3. The only
independent
Math 33B Practice Exam 2
Problem 1. Verify that y(t) = et and y(t) = tet are solutions to y 00 2y 0 + 2 y = 0.
Then prove they are linearly independent.
Problem 2. Find the general solution to 4y 00 +
Math 33B Exam 1 Solutions
Problem 1. Below is a list of statements. Decide which are true and which are false. On
the left of each, write TRUE or FALSE in capital letters. You must also write your
ans
Problem 1.
Answer: Plug into the equation to check they are solutions. Compute the Wronskian to
be W (t) = e2t , which is clearly non-zero.
Problem 2.
Answer: y(t) = A sin( 2t ), by solving characteri
HOMEWORK 10
Section 9.5 in the book: Exercises 8, 10, 14, 16, 22, 26, 28, 32.
Problem 1. For each of the following matrices, perform the following tasks:
a) classify the equilibrium point of the syst
HOMEWORK 9
Section 9.2 in the book: Exercises 30, 34, 36, 40, 42, 46, 50, 54, 58.
Section 9.3 in the book: Exercises 10, 14, 18, 20. (You may skip the parts of the
problems involving numerics.)
Prob
HOMEWORK 8
Section 9.1 in the book: Exercises 16, 22, 24, 26.
Section 9.2 in the book: Exercises 2, 6, 8, 12, 16, 20, 22, 26.
Problem 1. Initially, tank A contains 50 gallons of water in which 25 po
HOMEWORK 7
Section 4.6 in the book: Exercises 8, 12, 14.
Problem 1. Find the general solution to the equation
4x 4x + x = et/2
1 t2 .
Problem 2. Use the method of undetermined coecients (or the metho
HOMEWORK 5
Section 4.1 in the book: Exercises 22, 24, 26, 30.
Section 4.3 in the book: Exercises 24, 26, 32, 34, 36.
Problem 1. Show that for the dierential equation
y + y = 0,
(a) there are innitel
HOMEWORK 3
Section 2.6 in the book: Exercises 10, 12, 20, 26, 28, 30.
Problem 1. Show that the following dierential equation is exact
dy
y+x
= 0.
dx
Find an explicit formula for the solution y(x). Wh
HOMEWORK 6
Section 4.5 in the book: Exercises 18, 20, 22, 26, 38, 42, 44.
Problem 1. Consider the equation
(2 t)x + (2t 3)x tx + x = 0
for t < 2.
t
(a) Verify that 1 (t) = e is a solution.
(b) Look f
HOMEWORK 2
Section 2.4 in the book: Exercises 4, 16, 22, 28, 40.
Section 2.5 in the book: Exercises 8, 12.
Problem 1. (a) Let a R be a constant and let b1 : R R and b2 : R R be
two continuous functi