Description 
Resources 
Sample Problems 
Series and Power Series
 Recognize when a series is geometric
 Find the exact sum of a geometric series
 Classify a given series as convergent or divergent, using the Ratio Test
 Find the radius of convergence for a given power series
 Recognize and manipulate Taylor Series for famous functions

Class notes;
Trench Section 7.1.

HW01, Questions 14.
Trench 7.11, #3, 7.

ConstantCoefficient Linear ODE's
[ay" + by' + cy = 0]
 “Guess” a solution of suitable form; find the characteristic polynomial.
 Find the general solution for any combination of constants a,b,c
 Solve an initialvalue problem in which y(x_{0}), y'(x_{0})
are given
 Understand and exploit qualitative requirements on the solution functions y

Class notes;
Trench Section 5.2.

HW02, Question 4;
Trench Section 5.2, #117.

Linear ODE's of Euler Type
[ax^{2}y" + bxy' + cy = 0]
 “Guess” a solution of suitable form; find the characteristic polynomial.
 Find the general solution for any combination of constants a,b,c
 Solve an initialvalue problem in which y(x_{0}), y'(x_{0})
are given
 Understand and exploit qualitative requirements on the solution functions y
 Make suitable changes to handle problems where the singular point is not at x=0

Class notes;
Trench Section 7.4.

HW02, Question 6;
Trench Section 7.4, #118, 21.

Power Series Solutions for Linear ODE's
 Classify every point as either ordinary or singular
 For singular points, decide between regular and irregular
 Choose an expansion centre, based on given information
 Decide if the expansion centre is an ordinary point or a singular point
 For expansions around an ordinary point,
 Predict the form of two linearly independent series solutions
 Predict the minimum radius of convergence for those two solutions
 Find the recurrence relation for coefficients in the series solutions
 Determine the first five terms in both series solutions
 For expansions around a [regular] singular point
 Determine the approximating Eulerstyle equation
 Find the indicial equation and the exponents of singularity
 Predict the form of two linearly independent series solutions
 Predict the minimum radius of convergence for those two solutions
 Find the recurrence relation for coefficients in the series solutions
 Determine the first five terms in one of the series solutions
 If the exponents of singularity do not differ by an integer,
determine the first five terms in a second series solution

Class notes;
Trench Sections 7.2, 7.5, 7.6.

HW02, Questions 13;
Trench Section 7.2 #18, 1620;
Trench Section 7.5 #1425, 3346;
Trench Section 7.6 #1222, 2838.

Change of Independent Variable in ODE's
 Apply a given transformation relating x and t
to express a given ODE for unknown y=y(x)
into an equivalent ODE for unknown u=u(t)
 Recover solutions for the original problem involving y=y(x)
from solutions for the transformed problem involving u=u(t).

Class notes;
Trench's proof of Theorem 7.4.3.

HW03, Questions 13.

Eigenvalues and Eigenvectors, with ODE applications

Recognize the eigenvector/eigenvalue equation:
Av=λv.

Test a given vector for eigenvector status.
Find the corresponding eigenvalue, if appropriate.

Test a given constant for eigenvalue status.
Find the corresponding eigenvectors, if appropriate.

Know what special properties of eigenvalues and eigenvectors
are guaranteed by starting with a symmetric matrix.
Use these to determine eigenvectors in simple cases.

Task 1:
Given a collection of mutually orthogonal vectors,
find the coefficients in a linear combination that
matches some given vector.

Task 2:
Given the eigenvectors for a symmetric matrix A,
solve for x in the vectormatrix equation Ax=b.

Task 3:
Given the eigenvectors for a symmetric matrix A,
solve for u(t) in the vectormatrix differential
equation u'(t) = Au(t).

Class notes

HW04, #1, #3 (Part I)
HW05, #1 (Part I)

Eigenvalues and Eigenfunctions
[Lu=(p(x)u')'/r(x)+q(x)u/r(x)]

Recognize the key ingredients of a
general ODE eigenvalue problem:
ODE + interval + BC.

Recite the definition of an eigenvalue.

Recite the definition of an eigenfunction.

Test a given function for eigenfunction status (ODE+BC).
Find the corresponding eigenvalue, if appropriate.

Find all eigenfunctions in a given eigenvalue problem.
Possible methods:
 Recognize one of the Big Four eigenvalue problems
and recall the eigenfunctions from memory;
 Make a casebycase analysis on an ODE of
Euler or constantcoefficient type.
 Class notes;
Trench 11.1

HW04 #2, #3 (Part II(a))
Trench 11.1 #121

FSS, FCS, HPSS, HPCSTask 1: Coefficient Extraction

Choose an appropriate series form, guided by ODE/BC's.

Calculate coefficients in an eigenfunction expansion
of a given function f=f(x), either
 by coefficientmatching, or
 by integration

Plot the limit function by hand,
deducing behaviour outside the basic interval
from symmetry of the eigenfunctions and jumpaveraging.

Derive exact values for some series of constants by
combining the sketch with the eigenfunction series,
and evaluating at a particular value of x.

Generalize: Apply the coefficientextraction
procedure to any given family of functions with
a known orthogonality property.

Class notes;
Trench 11.2;
Trench 11.3.

HW04 #3 (Part II(b))
Trench 11.2 #1014, 19, 20.
Trench 11.3 #1, 7, 13, 17, 19, 22, 26, 28

FSS, FCS, HPSS, HPCSTask 2: Equation Solving
[Lu=f]
Confronted with an equation of the form
u''(x) + ku(x) = f(x)
for some given function f(x) and constant k,
...

Use the given BC's to select an appropriate family
of eigenfunctions.

Find an eigenfunctionseries representation for the
solution u(x).

Class notes

HW04 #3 (Part II(ce))

FSS, FCS, HPSS, HPCSTask 3: Dynamics
[du/dt=Au, etc.]
Confronted with a partial differential equation for function
u=u(x,t), ...

Use boundary data to determine a suitable family of
eigenfunctions

“Postulate”:
Write a template for the desired solution
in the form of an eigenfunction
series involving timevarying coefficients

“Initialize”:
Use initial data to find specific information about the
coefficients at t=0

“Propagate”:
Find and solve timeevolution ODE's for the coefficient
functions. Be ready for constantcoefficient linear dynamics
of either firstorder or secondorder.

Class notes

HW05 #1 (Part II, Part III)
Trench 12.1 #9, 10, 18, 20, 34, 37
Trench 12.2 #2, 5, 18, 35, 49, 55
NOTE: Textbook sections 12.1 and 12.2 contain important
material we have not yet discussed in class, and which
is not examinable. The skill required here is simply to
write seriesform solutions for problems of the type
shown here by following the steps listed at the left.
