January 4, 2015
Review of Second Order Linear Equations with
Constant Coefficients
Welcome to Math 3120!
Blackboard Learn:
https:/dalhousie.blackboard.com/
Please read the course outline on Blackboard
this week.
Chapter 3: Series Solutions to Differential
Orthogonality of Sines and Cosines
Sine and cosine functions are also orthogonal on
formulas are easy to get from the previous slide.
< f , g >=
Z
L
L
L x L. These
f (x)g(x) dx
Chapter 8: Fourier Series and Partial Differential
Equations
A partial differe
Fourier Series
In solving the heat equation with different boundary conditions in
examples 11, 12, and 13, we needed to equate a function f (x) with
an infinite linear combination of sines and cosines. We saw the
following three cases:
I
f (x) written as
Fourier Series
f (x)
Remark 6
Theorem 3: Convergence of Fourier Series
1
X
n=0
An cos
nx
L
+
1
X
n=1
Bn sin
nx
L
The infinite series above may or may not converge. Furthermore, if it
converges it may not converge to f . The means that f (x) is on the
Some Linear Algebra before Chapter 9
Theorem 4: Let A be an nxn symmetric (A = A> ) matrix with real
entries. Then
I
All the eigenvalues of A are real
I
There is an orthogonal basis for Rn consisting of the
eigenvectors of A .
Remark 7
As mentioned above,
Theorem 5: Regular Sturm Liouville Theorem
The following results apply to the regular SL problem defined in (9):
I All the eigenvalues
are real.
I
Theorem 5 Continued . . .
I
Z
There exists an infinite number of eigenvalues
1
<
2
< . <
n
<
n+1
< .
I
I
I
C
Theorem 2: Frobenius Series Solutions
Theorem 2 continued . . .
Suppose x = 0 is a regular singular point of (5). This means
xP(x) =
x 2 Q(x) =
1
X
n=0
1
X
(b) If r1 r2 is not zero and not a positive integer then there exists a
second linearly independent
Theorem 1: Solutions Near an Ordinary Point
Example 4 (Chebyshev equation)
Suppose x = a is an ordinary point of
00
0
y + P(x)y + Q(x)y = 0.
(3)
Then the differential equation in (3) has two linearly independent
solutions each of the form
y (x) =
1
X
cn (
3.2 Series Solutions Near Ordinary Points
Example 3
xy 00 + sin (x)y 0 + x 2 y = 0
A(x)y 00 + B(x)y 0 + C(x)y = 0
can be rewritten as
(2)
Definition 4 (ordinary point)
A point x = a is called an ordinary point of (2) if both P(x) and Q(x)
are analytic at
Some examples of common power series:
Analytic Functions
ex =
Definition 3 (Analytic Function)
sin x =
A function f (x) is analytic at a point x = a if it can be represented by
a power series in (x a) with R > 0. This means the power series
converges to f
Review of Second Order Linear Equations with
Constant Coefficients
Chapter 3: Series Solutions to Differential Equations
A(x)y 00 + B(x)y 0 + C(x)y = 0
(1)
I
Linear
2nd Order
I
Homogeneous
I
Variable coefficients
I
In general, cant find solutions of (1) i
Chapter 7: Nonlinear Systems
Definition019 (Autonomous
System of DEs)
1
x1 (t)
Bx2 (t)C
B
C
Let X (t) = B . C, then
@ . A
xn (t)
d
X = F (X )
dt
is called an autonomous system of differential equations. An initial
condition is usually given and denoted as
Chapter 9: Sturm-Liouville Eigenvalue Problems
It turns out the eigenvalue problems that we kept seeing before, such
as
y 00 + y = 0
0 < x < L,
y (0) = y (L) = 0,
are special cases of a more general theory called Sturm-Liouville
theory.
Sturm-Liouville Op
Translated Series Solutions and Initial Conditions
Consider the initial value problem
We again consider this ODE with analytic coefficients,
y 00 + P(x)y 0 + Q(x)y = 0
y 0 (x0 ) = 1
y (x0 ) = 0
3.3 Regular Singular Points
A(x)y 00 + B(x)y 0 + C(x)y = 0.
T
Some examples of common power series:
Analytic Functions
ex =
Definition 3 (Analytic Function)
sin x =
A function f (x) is analytic at a point x = a if it can be represented by
a power series in (x a) with R > 0. This means the power series
converges to f
Theorem 2: Frobenius Series Solutions
Theorem 2 continued . . .
Suppose x = 0 is a regular singular point of (5). This means
xP(x) =
x 2 Q(x) =
1
X
n=0
1
X
(b) If r1 r2 is not zero and not a positive integer then there exists a
second linearly independent
Theorem 1: Solutions Near an Ordinary Point
Example 4 (Chebyshev equation)
Suppose x = a is an ordinary point of
00
0
y + P(x)y + Q(x)y = 0.
(3)
Then the differential equation in (3) has two linearly independent
solutions each of the form
y (x) =
1
X
cn (
Review of Second Order Linear Equations with
Constant Coefficients
Chapter 3: Series Solutions to Differential Equations
0
(1)
Linear
2nd Order
Homogeneous
Variable coefficients
In general, cant find solutions of (1) in terms of elementary
functions (sin
3.2 Series Solutions Near Ordinary Points
Example 3
xy 00 + sin (x)y 0 + x 2 y = 0
A(x)y 00 + B(x)y 0 + C(x)y = 0
can be rewritten as
(2)
Definition 4 (ordinary point)
A point x = a is called an ordinary point of (2) if both P(x) and Q(x)
are analytic at
Piecewise Continuous and Piecewise Smooth
Some Graphs of Piecewise Smooth Functions
Definition 10 (Piecewise Continuous)
A function f (x) is piecewise continuous on an interval [a, b] provided
that [a, b] can be subdivided into finitely many subintervals
Fourier Sine and Fourier Cosine Series
Fourier Sine and Fourier Cosine Series
Definition 13 (Even and Odd Functions)
I
A function f (x) is even if f ( x) = f (x). A function f (x) is odd if
f ( x) = f (x).
A Fourier series for an even function has just co
Review of some Linear Algebra before starting
Chapter 8
0
1
0 1
u1
v1
Bu2 C
B v2 C
B C
B C
Recall that two vectors u = B . C and v = B . C 2 Rn are
@ . A
@ . A
un
vn
orthogonal if the dot product u v = 0. The dot product is defined as
u v = u1 v1 + u2 v2
Heat Equation
Solving Heat Equation with Separation of Variables
Consider a bar of length L of uniform cross section
Let u(x, t) be the temperature of the bar at some point along the x
axis and some point in time.
u(x, t) satisfies the heat equation (see
Translated Series Solutions and Initial Conditions
Consider the initial value problem
We again consider this ODE with analytic coefficients,
y 00 + P(x)y 0 + Q(x)y = 0
y 0 (x0 ) = 1
y (x0 ) = 0
3.3 Regular Singular Points
A(x)y 00 + B(x)y 0 + C(x)y = 0.
T
Math 3120 Dierential Equations II
Homework #3 Solutions
1. For some types of infection, there is a latent period in which an individual is infected, but
not yet infectious. We can call this the exposed stage. We let a be the rate at which exposed
individu
Math 3120 - Dierential Equations II
1
Course Outline
Text: Elementary Dierential Equations with Boundary Value Problems, C. Edwards and D. Penney
In this course, we will consider some of the more advanced topics in dierential equations. The key topics we
Math 3120 Dierential Equations II
Homework #3 Due Monday March 18
1. For some types of infection, there is a latent period in which an individual is infected, but
not yet infectious. We can call this the exposed stage. We let a be the rate at which expose
Math 3120 Dierential Equations II
Homework #2 Solutions
1. Find two linearly independent Frobenius series solutions, about x = 0, to the dierential
equation
(x + 2)x2 y xy + (1 + x)y = 0 .
We rst note that x = 0 is a regular singular point.
1
1
= ,
x0 x +
Math 3120 Dierential Equations II
Homework #2 Due Friday February 15
1. Find two linearly independent Frobenius series solutions, about x = 0, to the dierential
equation
(x + 2)x2 y xy + (1 + x)y = 0 .
2. Consider the dierential Equation
xy + 3y xy = 0 .
Math 3120 Dierential Equations II
Homework #1 Solutions
1. Find the radius of convergence about the given point, x0 , for which the following dierential
equations with initial conditions given at x0 are guaranteed to have a unique solution analytic
in the