Math 138A, Fall 2015, Term Test 1
Ordinary Differential Equations
Date: Thursday, October 1
Time: 1:302:45 pm.
Lecture Section: 002
Instructor: Matthew Johnston
Surname (Family Name):
Given Name:
SJ SU Student ID Number:
FOR. EXAMINERS USE ONLY
Instructio
Math 133A Midterm#1a
Wednesday, March 4, 2015
10:22 PM
New Section 1 Page 1
New Section 1 Page 2
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Math 133A
Midterm #3
D. Goldston
NAME
Problem
Points
1
15
2
15
3
20
4
25
5
25
Total
100
Score
Nov. 20, 2013
Show all work. You MUST show steps to get credit. The correct answer alone
is only worth 2 points.
1. Find all the singular points of the following
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mm.
Show all work. You MUST Show steps to get credit. The correct answer alone
is only worth 1 point.
1. Find the general solution of
(30 points, 6 each)
a)y+y*12y=0 )
)
2. Find the general solution of
8 each
(24 points
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San Jose State University
Math 133A, Fall 2015
Quiz 1 Solution
Consider the dierential equation
y = y 3 + 3y 2 10y.
(a) What are the equilibrium solution?
(b) For which values of y is y(t) increasing?
Solution: (a) Let
f (y) = y 3 + 3y 2 10y.
Factoring w
San Jose State University
Math 133A, Fall 2015
Quiz 2 Solution
Find the general solution of the dierential equation
1
dy
= 2
.
dt
t y + t2 + y + 1
Solution: Since
1
1
= 2
,
t2 y + t2 + y + 1
(t + 1)(y + 1)
the equation is separable. Since the right-hand
San Jose State University
Math 133A, Fall 2015
Quiz 4 solution
Sketch the phase line of the given dierential equation and classify the equilibria into sinks,
sources, and nodes.
w = (w3 w2 )(w2 + 2w + 1).
Solution: Denote the right-hand side of the equat
1. (25 points) Consider the dierential equation
dy
= (y 2 y 2) log(1 + y 2 ),
dt
where log denotes the natural logarithm.
(a) Draw the phase line.
(b) Classify all equilibria.
(c) Approximately sketch the solution satisfying the initial condition y(0) = 1
San Jose State University
Math 133A, Fall 2015
Practice ALWAYS-SOMETIMES-NEVER Questions
You should also be able to answer questions of the following type:
For each of the following statements, determine if the conclusion ALWAYS follows from the
assumpti
San Jose State University
Math 133A, Fall 2015
Solution to ALWAYS-SOMETIMES-NEVER Questions
You should also be able to answer questions of the following type:
For each of the following statements, determine if the conclusion ALWAYS follows from the
assum
FALL 2015
HOMEWORK 3 SOLUTIONS
Section 1.5
We will use the following formula for the general solution of y + p(t)y = q(t):
y=
where (t) = exp
1
(t)
(t)q(t) dt + C ,
p(t) dt .
1. Since p(t) = 1, we have (t) = et so the general solution is
y = et (e2t + C).
FALL 2015
HOMEWORK 5 SOLUTIONS
Section 1.8
3
1. Let f (y) = yey ln(1 + y 2 ). Then by the Product and Chain Rules, we have
3
3
f (y) = ey ln(1 + y 2 ) + y 3y 2 ey ln(1 + y 2 ) + yey
3
2y
,
1 + y2
so f (0) = 0. Therefore, the Linearization Theorem is incon