UBC Math 257(921): Midterm Test
Thursday 19 June 2014, 14:00-15:50, LSK 200
Examinable Topics

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: Avv.
  • 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.

Rules

Equipment Required: Not Permitted: Integrity (from our Course Outline):
On both homework and tests, the highest possible standard of academic integrity is expected and will be enforced. In short, students are expected to submit work that represents their own current thoughts and understanding. For details, review the UBC Calendar's sections on Academic Honesty and Standards and Academic Misconduct.

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