Ch 7.7: Fundamental Matrices
Suppose that x(1)(t), x(n)(t) form a fundamental set of solutions f
or x' = P(t)x on
<t< .
The matrix
x1(1) (t )
x1( n ) (t )
(t )
,
xn(1) (t )
xn( n ) (t )
whose columns are x(1)(t), x(n)(t), is a fundamental matrix for the
Math 201 Lecture 24: Power Series Method: Analytic Coefficients
Mar. 09, 2012
Many examples here are taken from the textbook. The first number in () refers to the problem number
in the UA Custom edition, the second number in () refers to the problem numbe
Math 201 Lecture 18: Convolution
Feb. 17, 2012
Many examples here are taken from the textbook. The first number in () refers to the problem number
in the UA Custom edition, the second number in () refers to the problem number in the 8th edition.
0. Review
Math 201 Lecture Notes Winter 2014
5
Vincent Bouchard
Power series solutions to ODEs (section 8)
Learning outcomes (section 5)
You should be able to solve linear ODEs by using power series at an ordinary point x = x0 ,
and know how to determine whether x
Math 201 Lecture 23: Power Series Method for Equations with Polynomial Coefficients
Mar. 07, 2012
Many examples here are taken from the textbook. The first number in () refers to the problem number
in the UA Custom edition, the second number in () refers
Math 201 Lecture 37: Review of 7.77.9
Apr. 13, 2012
Example 1. Solve the problem
ex y y = sin x,
y(0) = 1, y (0) = 0
(1)
up to first four nonzero terms.
Solution. Write
X
y=
an xn = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 +
(2)
n=0
We write the first 5 terms b
Math 201 Lecture Notes Winter 2014
6
Vincent Bouchard
Partial differential equations (section 10)
Learning outcomes (section 6)
You should be able to solve simple initial-boundary problems (such as the one-dimensional
heat equation and the string equatio
Math 201 Winter 2012 Homework 6 (8.2 8.4)
Problem 1. Determine the convergence set of the given power series.
P 3
n
a)
n=1 n8 (x 2) ,
P
n 3n
b)
n=0 2 x .
Problem 2. Let f (x) and g(x) be defined by the following two power series:
X
(1)n n
x ,
n
X
n=1
n=0
Math 201 Lecture Notes Winter 2014
2
Vincent Bouchard
First-order differential equations (section 2)
Remark. The section numbers in the titles refer to the textbook Fundamentals of dierential equations by Nagle, Sna and Snider.
Learning outcomes (section
Chapter 2. First order Differential equations
Section 2.1 Linear Equations; Method of Integrating
Factors
A linear first order ODE has the general form
!
dy
= f (t , y )
dt
where f is linear in y. Examples include equations with constant
coefficients, suc
Dierential Equations
Spring 2016
Jung Eun Kim
Textbook: Elementary Differential Equations and Boundary Value
Problems, Boyce and DiPrima (10th ed.)
Lecturer: Kim, Jung Eun
Email: [email protected] or [email protected]
Office Hours: M, W 14:30 15:50
Ch 4.1: Higher Order Linear ODEs: General Theory
n-th order linear ODE
An n-th order linear ODE has the general form
Pn (t)y (n) (t)+Pn1 (t)y (n1) (t)+ +P2 (t)y 00 (t)+P1 (t)y 0 (t)+P0 (t)y (t) = G(t).
Assume that P0 , , Pn , and G are continuous real-val
Math 201 Prerequisites for 10.3,10.4
1. Trignometric Identities
The following trignometric identities will be used again and again in calculation of Fourier/Fourier
Cosine/Fourier Sine series:
1
[cos ( + ) + cos ( )];
2
1
[cos ( ) cos ( + )];
sin sin =
2
Part A Fluid Dynamics & Waves
Draft date: 21 January 2014 21
2 Two-dimensional
incompressible irrotational flow
2.1
Velocity potential and streamfunction
We now focus on purely two-dimensional flows, in which the velocity takes the form
u(x, y, t) = u(x,
2.016 Hydrodynamics
Reading #4
2.016 Hydrodynamics
Prof. A.H. Techet
Potential Flow Theory
When a flow is both frictionless and irrotational, pleasant things happen. F.M.
White, Fluid Mechanics 4th ed.
We can treat external flows around bodies as invicid
Ch 6.1: Definition of Laplace Transform
I
Many practical engineering problems involve mechanical or
electrical systems acted upon by discontinuous or
impulsive forcing terms.
I
For such problems the methods described in Chapter 3 are
difficult to apply.
I
Section 3.1: 2nd Order Linear Homogeneous
Equations-Constant Coefficients
I
A second order ordinary differential equation has the general
form
y 00 = f (t, y , y 0 )
where f is some given function.
I
This equation is said to be linear if f is linear in y
Ch 5.1: Review of Power Series
I
Finding the general solution of a linear differential equation
depends on determining a fundamental set of solutions of the
homogeneous equation.
I
So far, we have a systematic procedure for constructing
fundamental soluti
Section 2.6 Exact Equations and Integrating Factors
Exact equation
Consider a first order ODE of the form
M(x, y ) + N(x, y )y 0 = 0.
This equation is called exact, if there is a function such that
x (x, y ) = M(x, y ),
y (x, y ) = N(x, y ).
Exact equatio
Math 201 Lecture 20: Review of Chapter 4 and 8.5
Feb. 29, 2012
Many examples here are taken from the textbook. The first number in () refers to the problem number
in the UA Custom edition, the second number in () refers to the problem number in the 8th ed
Math 201 Lecture Notes Winter 2014
3
Vincent Bouchard
Linear second-order ODEs (section 4)
Learning outcomes (section 3)
You should be able to understand the distinction between equations with constant coecients vs equations with variable coecients, and
Math 201 Final Review Problem Set 2 Solution
Multiple Choice
1. Taking Laplace transform of the equation:
Lcfw_y = s2 Y s y(0) y (0) = s2 Y s 1;
(1)
Lcfw_y = s Y y(0) = s Y 1.
(2)
Lcfw_(t ) = es.
(3)
The transformed equation then reads
which simplifies
Q&A 02
Jan. 1620 , 2012
1. Q. When solving
y + 2 y 8 y = 0,
y(0) = 3, y (0) = 12
(1)
you write the general solution as
C1 e4t + C2 e2t ,
(2)
C1 e2t + C2 e4t.
(3)
but I write it as
Does it matter?
A. No. Equivalently, the question is: When solving the char
Collection of Common Mistakes 03: Constant Coefficients, Undetermined Coefficients
Jan. 27, 2012
1. Spot Mistakes
There may be more than one mistakes in one single solution.
Problem 1. Solve y + 2 y + 4 y = 0.
Solution 1.
r2 + 2 r + 4 = 0
Solution 2.
r2 +
MATH 201 ASSIGNMENT 9.
Problem 1.
Find a formal solution to the initial-boundary value problem:
u
2u
= 2 2 , 0 < x < , t > 0,
t
x
u(0, t) = 5,
u(, t) = 10, t > 0,
u(x, 0)
sin 3x sin 5x,
=
0 < x < .
Problem 2.
Find a formal solution to the initial-boundary
Homework 8
1
Exercise 1
Compute the Fourier series on the specified interval of the following functions
and sketch a picture of the function it converges to. Compare it to the original
function.
0
if < x < 2
1 if 2 < x < 0
f (x) =
(1)
1
if 0 < x < 2
0
if
Math 201 Prerequisites for Week 2 (2.6, 4.2, 4.3)
1. Complex Numbers
A complex number is of the form
a+bi
(1)
where a, b are real numbers, and i is a symbol, representing the imaginary number which solves
x2 = 1.
For us, the most relavent property of comp