Math 22B Solutions Homework 2 Spring 2008
Section 2.1 16. y + 2 y = t
cos t , t2
with y() = 0 and t = 0
R
2 dt t
Solution Let (t) = e equation by (t), we get:
= et . If we multiply both sides of the given 2 = cos t t
2
t2 (y) +
(t2 y) = cos t
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2.5. Autonomous Equations
MAT 22B 001
14 APR 2014
2.5 Autonomous Equations
A. Equilibria of Autonomous Equations
A constant solution y(x) = y to an ODE is called an equilibrium.
We study the equilibria of autonomous equations of the form
y = g(y).
Observa
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MAT 22B 001
1.1. HW Solutions
1.1
14 APR 2014
HW Solutions
#7. There is no unique solution to this problem; we look for an equation of the form
y = ay + b. For this equation to have an equilibrium y = 3, we need b = 3a, so
we have y = ay 3a. If we take a
MAT 22B 001
2.5. HW Solutions
2.5
18 APR 2014
HW Solutions
#2. dy/dt = ay + by 2 , a > 0, b > 0, < y0 <
Let f (y) = ay + by 2 ; then f (y) = a + 2by. There are two equilibria, y = a/b, 0.
f (a/b) = a < 0 y = a/b is stable. f (0) = a > 0 y = 0 is unstable
MAT 22B 001
1.3. HW Solutions
1.3
15 APR 2014
HW Solutions
#2. Second-order non-linear ODE (the coecient in front of y makes it non-linear).
#4. First-order non-linear ODE (the presence of y 2 term makes it non-linear).
#6. Third-order linear ODE (the coe
1.2. HW Solutions
MAT 22B 001
1.2
15 APR 2014
HW Solutions
#3. y = ay + b with a > 0, b > 0.
(a) (ay + b)1 dy = dt ln |ay+b| = t + c1 ln | ay + b| = at + c2
a
| ay + b| = eat+c2 = ec2 eat ay + b = c3 eat y = Ceat + (b/a) with C = 0;
however, the equilibr
MAT 22B 001
3.2. Solutions of Linear Equations
9 APR 2014
3.2 Solutions of Linear Equations
A. Existence and Uniqueness
Thoerem 3.2.1 (EUT)
Let the functions p, q, g be continuous on an open interval
1 < t < 2, containing the point t0; then, on the interv
MAT 22B 001
1.2. Solutions of Dierential Equations
2 APR 2014
1.2 Solutions of Dierential Equations
A. Solutions
We want to understand the unknown function in the given dierential
equation; in particular, we shall address the following questions.
Solutio
MAT 22B 001
2.1. Linear Equations
4 APR 2014
2.1 Linear Equations
A. First-Order Linear Equations
A rst-order linear ODE is an ODE that can be written as
y + p(x) y = q(x).
When p(x), q(x) are constant functions, the equation specializes to
y ay = b or eq
Lecture 7 Wednesday October 5, 2016
There is great intrinsic value in maths: Nothing is as straightforward and honest.
Mathematics is the door and key to the sciences.
Roger Bacon
Mathematics is the science of what is clear by itself.
Carl Gustav Jacob Ja
Math 22B: Differential Equations (Fall 2016)
Version 1
INSTRUCTORS CONTACT INFORMATION
Instructor:
William E. Tavernetti
Email:
[email protected]
Office:
MSB 1103
Tutoring Hours:
MWF 1 - 3+ at my office MSB 1103 (the + means I will stay late if we
Lecture 5 Friday Sept 30, 2016
"A mind is a fire to be kindled, not a vessel to be lled. - Plutarch
Nothing was ever achieved without enthusiasm." - Emerson
Schedule for Lecture
' Warm-up
Finish example from last class (2.2)
Second interval of validity
Homework 1: Math 22B - Tavernetti, Fall 2016
Due Wednesday, Sept 28 in class
(20 points)
Instructions : Solve all problems. Print out your solutions when computer results are asked for
(do not include the dfield8.m code), work neatly, label your plots, sh
Homework 2: Math 22B - Tavernetti, Fall 2016
Due Wednesday, Oct 5 in class
(20 points)
Instructions : Solve all problems. Print out your solutions when computer results are asked for
(do not include the dfield8.m code), work neatly, label your plots, show
Lecture 2 Friday Sept 23, 2016
"Geometry existed before creation." Plato
"Mathematics takes us still further from what is human, into the region of absolute necessity, to
which not only the actual world, but every possible world, must conform." Bertrand R
Lecture 3 Monday Sept 26, 2016
"The essence of mathematics is in its eedom." -~ Georg Cantor
Calm-Hilliard / lnpainting / Image Restoration
Schedule for Lani/J"
' Warm-up
' Modeling (1.1-1.2)
0 Radioactive decay
Verifying Solutions (1.3)
' Preview of 2