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 of , sin(t + ) = cos(t), which is a scalar
multiple of c

Math 33B Practice Exam 3 Solutions
Problem 1.
(a) y = cex
(b)
y2
2
2 /2
+ cos x = 12 .
(c) y = cex + 2
(d)
1
(x2 +1)3/2
+2
(e) y = c x2 .
(f) y = cx3/2 1.
(g)
1
6
ln
x2 xy+3y 2
x2
1
tan1
+
y
6y
1
x
11
3 11
=c
ln(x)
4
y
(h) x = Ce 2 (ln( x )+ln( x +2) .
Pr

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 found to be 2 + 3 + 2, with roots 1 and 2.
Hence we have a

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 eigenvector is (1, 0, 0), so that the geometric multipl

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 eigenvector is (1, 0, 1), so that the geometric multip

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 + y = 0. Then find the particular solution
that satisfie

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
answer (TRUE or FALSE in capital letters) on the front pag

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 characteristic equation and absorbing the second
).
piece into a

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 system x = Ax based on the position of
(T, D) in the trace

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.)
Problem 1. (CayleyHamilton Theorem) Let
A=
a
c
b
d
be a mat

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 pounds of
salt are dissolved, while a second tank B conta

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 method of annihilators) to nd the general solution to the eq

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 innitely many solutions obeying y(0) = y() = 0;
(b) there is e

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). What is the interval of existence of
the solution?
Proble

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 for a solution of the form 2 (t) = v(t)1 (t). Plug this

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 functions such that b1 (t) b2 (t) for all t 0. Consider the l