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
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
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 constan
MAT 22B 001
2.2. Separable Equations
7 APR 2014
2.2 Separable Equations
A. First-Order Separable Equations
A rst-order separable ODE is an ODE that can be written as
h(y)
dy
= g(x).
dx
When g(x) = 1 a
MAT 22B 001
3.1. Linear Homogeneous Equations I
24 APR 2014
3.1 Linear Homogeneous Equations I
A. Second Order Linear Equations
A second-order linear ODE is an ODE that can be written as
y + p(t) y +
MAT 22B 001
2.8. Existence and Uniquness
14 APR 2014
2.8 Existence and Uniquness
A. Restatement of EUT
Whenever we consider an initial value problem y = f (t, y), y(t0) = y0,
we can apply a simple pre
MAT 22B 001
2.3. Modeling with First-Order Equations
9 APR 2014
2.3 Modeling with First-Order Equations
A. One-Compartment Mixing Models
A one-compartment mixing model concerns the amount of a
chemica
MAT 22B 001
1.1. Utility of Dierential Equations
31 MAR 2014
1.1 Utility of Dierential Equations
A. Dierential Equations
A dierential equation is an equation involving variables, an unknown function i
MAT 22B 001
2.4. Linear vs. Non-Linear
14 APR 2014
2.4 Linear vs. Non-Linear
A. Existence and Uniqueness of Solutions
The existence and uniqueness theorem (EUT) stipulates that
there exists a unique s
MAT 22B 001
1.3. Classication of Dierential Equations
4 APR 2014
1.3 Classication of Dierential Equations
A. Ordinary and Partial
An ordinary dierential equation is a dierential equation with
the unk
Math 22B
February 3, 2017
Review of Linear Algebra
1
1.1
Eigenvalues and eigenvectors of a matrix
Definitions
Let A denote a n n matrix acting on vectors in the vector space V (think of V as either Rn
The Two-Dimensional Laplacian
in
Cartesian & Polar Coordinates
The two-dimensional Laplacian in Cartesian coordinates is
2
2
= 2 + 2
x
y
and the two-dimensional Helmholtz equation is
u + k 2 u = 0.
We
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 particu
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
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 =
7 1 Msxr 005;,
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DC" = H DC; DC. at) KW WI so WE
Ear
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 equilibri
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Craig A. Tracy
Winter 2017
Mathematics 22B
The Final Examination, March 24, 2017
Instructions: Work all four problems in your bluebook. Only the bluebook will be collected.
? Useful Information You Ma
Work and Heat Transfer
By: S K Mondal
Chapter 3
A
bar
p
50
B
pV1.3 = c
C
0.2
0.4
0.8
V1 m3
Area under BC
p V p2 V2
= 1 1
n 1
50 105 0.4 20.31 105 0.8
W
=
1.3 1
= 1.251MJ
Here
pB = pB = 50 bar = 50 10
Work and Heat Transfer
By: S K Mondal
Solution:
Chapter 3
Change of volume = A L
d 2
L
4
0.4 2
=
0.485 m3
4
= 0.061 m3
=
As piston moves against constant atmospheric pressure then work done = pV
=
Temperature
By: S K Mondal
gives the ratio of Sb.p. : H2Ob.p. On a gas thermometer operating at zero
gas pressure, i.e., an ideal gas thermometer. What is the boiling point of
sulphur on the gas scale