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 1-4.
Trench 7.11, #3, 7.
|
Constant-Coefficient 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 initial-value problem in which y(x0), y'(x0)
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, #1-17.
|
Linear ODE's of Euler Type
[ax2y" + 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 initial-value problem in which y(x0), y'(x0)
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, #1-18, 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 Euler-style 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 1-3;
Trench Section 7.2 #1-8, 16-20;
Trench Section 7.5 #14-25, 33-46;
Trench Section 7.6 #12-22, 28-38.
|
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 1-3.
|
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 vector-matrix equation Ax=b.
-
Task 3:
Given the eigenvectors for a symmetric matrix A,
solve for u(t) in the vector-matrix 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 case-by-case analysis on an ODE of
Euler or constant-coefficient type.
| Class notes;
Trench 11.1
|
HW04 #2, #3 (Part II(a))
Trench 11.1 #1-21
|
FSS, FCS, HPSS, HPCS--Task 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 coefficient-matching, or
- by integration
-
Plot the limit function by hand,
deducing behaviour outside the basic interval
from symmetry of the eigenfunctions and jump-averaging.
-
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 coefficient-extraction
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 #10-14, 19, 20.
Trench 11.3 #1, 7, 13, 17, 19, 22, 26, 28
|
FSS, FCS, HPSS, HPCS--Task 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 eigenfunction-series representation for the
solution u(x).
|
Class notes
|
HW04 #3 (Part II(c-e))
|
FSS, FCS, HPSS, HPCS--Task 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 time-varying coefficients
-
“Initialize”:
Use initial data to find specific information about the
coefficients at t=0
-
“Propagate”:
Find and solve time-evolution ODE's for the coefficient
functions. Be ready for constant-coefficient linear dynamics
of either first-order or second-order.
|
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 series-form solutions for problems of the type
shown here by following the steps listed at the left.
|