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
Math 33B-3, Spring 2015
Quiz 1
April 8, 2015
Name:
Student ID:
Section (circle one): 3A 3B
Show all work clearly and in order, and box your final answers. Justify your answers algebraically
and find explicit solutions whenever possible. There are no calcu
Math 33B-3, Spring 2015
Practice Midterm 1 Solutions
1.
This equation is homogeneous of degree 2. Using the substitution y = xv,
(x2 + x2 v 2 )dx + (x2 x2 v)(xdv + vdx) = 0
this simplifies to the separable
1
v1
dx =
dv
x
v+1
Use long division to integrate
Math 33B-3, Spring 2015
Quiz 3
May 12, 2015
Name:
Student ID:
(you must be enrolled in Section 3A)
Show all work clearly and in order, and box your final answers. Justify your answers algebraically
and find explicit solutions whenever possible. There are
Math 33B-3, Spring 2015
Quiz 3
May 14, 2015
Name:
Student ID:
(you must be enrolled in Section 3B )
Show all work clearly and in order, and box your final answers. Justify your answers algebraically
and find explicit solutions whenever possible. There are
Math 33B-3, Spring 2015
Quiz 4
May 27, 2015
Name:
Student ID:
Section (circle one): 3A 3B
Show all work clearly and in order, and box your final answers. Justify your answers algebraically
and find explicit solutions whenever possible. There are no calcul
Math 33B-3, Spring 2015
Quiz 5
June 3, 2015
Name:
Student ID:
Section (circle one): 3A 3B
Show all work clearly and in order, and box your final answers. Justify your answers algebraically
and find explicit solutions whenever possible. There are no calcul
Math 33B-3, Spring 2015
Quiz 2
May 1, 2015
Name:
Student ID:
Section (circle one): 3A 3B
Show all work clearly and in order, and box your final answers. Justify your answers algebraically
and find explicit solutions whenever possible. There are no calcula
§t0¥10n
MIDTERM 1
Math 33B
1/25/2010 Name:
Signature: m g
Read all of the following information before starting the exam:
0 Check your exam to make sure all pages are prmnt.
0 Show all work, clearly and in order, if you want to get full credit. I reserve
Math 33B
Instructor: Erkao Bao
Email: bao at math dot ucla dot edu
Office: MS 7340
Tentative Office hours: 9:55-10:55 MW or by appointment
Math help: Student Math Center 9am-3pm (M-TR) in MS3974. Detailed schedule can be
found at: www.math.ucla.edu/ugrad/
Math 33B Practice Exam 1 Answer Key
Problem 1. Show that sin(2x) and cos(2x) are solutions to the dierential equation
y + 4y = 0.
Answer.
Plug the functions into the dierential equation.
Problem 2. Find the general solution to the dierential equation
y =
Math 33B Quiz 4A
Name:
SID:
Problem 1. A 2 kg mass stretches a spring 1 m. Suppose the mass is set in motion from its
equilibrium position with a velocity of 2 m/s downwards. If there is no damping, and giving
the downward direction the negative orientati
Math 33B Quiz 3B
Name:
SID:
Problem 1. Find the solution of the initial value problem,
y + 10y + 25y = 0,
y(0) = 2,
y (0) = 1.
Solution: The characteristic equation is
2 + 10 + 25 = ( + 5)2 = 0.
We have a repeated root = 5. Therefore, the general solution
Math 33B Quiz 3A
Name:
SID:
Problem 1. Find the solution of the initial value problem
y y 2y = 0,
y(0) = 1,
y (0) = 2
Solution:
The characteristic equation is
2 2 = ( 2)( + 1) = 0.
So the roots are = 2 and = 1. Then the general solution is
y(t) = C1 e2t +
Math 33B Quiz 2B
Name:
SID:
Problem 1. Show that
f (t) =
0
t4
for t < 0
for t 0
is dierentiable for all values of t and solves the initial value problem ty = 4y, where
y(0) = 0. Find a second solution and explain why this does not contradict the uniquenes
Math 33B Quiz 2A
Name:
SID:
Problem 1. Solve the homogeneous dierential equation
y + 2xey/x dx xdy = 0
Solution: The equation is homogeneous of degree 1. For a homogeneous equation, use the
substitution y = xv. The equation becomes
0 =(vx + 2xev )dx xd(xv
Math 33B Quiz 1A
Name:
SID:
Problem 1. Find the solution of the initial value problem
xy + 2y = sin x,
y
2
= 0,
and determine the interval of existence.
Solution: Assuming x = 0, we can rewrite the equation as
2
sin x
y = y+
.
x
x
The integration factor i