Differential Equations - Study Guide - Final Exam

Differential Equations

Study Guide - Final Exam

Nature of the exam

The final assessment is cumulative and tests whether you have learned the concepts and derivations in the packets, and the techniques used in the exercises and problems. There will not be an intentional emphasis on material after the midterm, rather all packets are equally (likely to be) represented. As with the midterm, the final does not emphasize creative problem solving or rigorous argumentation, both of which are examined in your homework problem sets.

The format will consist of a single part with a simple sequence of problems on topics that progress through the packets, roughly in order. For each problem, the first 1-3 sub-questions will focus on memory of definitions, formulas, or derivations, while the last 1-2 sub-questions will ask for a calculation or solution to a small problem. Many such problems will be drawn from the Exercises and Problems of the course, but that is not guaranteed for all of them.

Three derivations are guaranteed to be chosen from among the following derivations: (i) solution to separable equations (not the mnemonic rule!) (ii) solution to first-order linear equations (all steps, not final formula!) (iii) solution to exact equations (not with integrating factor) (iv) solution to homogeneous equations (v) order linear homogeneous constant coefficient ( solutions) (vi) derivation of orthogonality of Fourier terms (‘modes’)

Study tips

Create your own set of complete notes

• Very condensed, summary level, comprehensive coverage, as if for a cheat-sheet.
• Write out all definitions and formulas, and illustrate them with example calculations.
• Include boxes containing longer derivations, examples, and problem solutions, placed under the theory elements that they rely on.
• Spend time studying these notes to memorize formulas and learn everything! You should spend at least as much time studying the notes as you did creating them.
• While studying it is probably helpful to write a second copy of the notes in more compressed form (only 30% the length of your first notes); compressing the notes is a helpful activity, and the compressed version is easier to memorize.
• To check your memory, give yourself short prompts, and try to unpack full details on paper or aloud, then check against your notes.

Meet with your Group members to study for the final! Review each other’s notes and quiz each other on concepts and problems.

Examinable content, per Packet

Items on the Midterm Study Guide that are not examinable for the Final are crossed out.

Packets 02-03

General derivations of methods to memorize in the abstract:

• Separable equations
  • must memorize a ‘correct’ derivation, not the mnemonic rule
  • you can use the mnemonic rule to solve separable equations on exams
• First-order linear equations
  • derivation steps, not just final solution formula
• Exact equations
  • without integrating factors
  • you can assume that a curl-free field is a gradient, or the algebraic implication of this fact
  • you can provide either the geometric derivation (using vector fields) or the algebraic one (using chain rule)
• Homogeneous equations (conversion to separable)
• Linear quotients (conversion to homogeneous)
• Reduction of order (conversion to first order, missing or missing )

Derivations not to memorize:

• Integrating factors for exact equations
• Bernoulli equations
• Linear substitutions
• Swapping roles

Additionally, for Packets 02-03:

• Can skip the “examples” of ODEs in first section of Packet 02
• For the midterm, can skip concept of “linear differential operator”
• Know how to solve the ODE types whose derivations are not required (integrating factors, Bernoulli, linear substitutions, linear quotients, swapping)
• Be able to identify all forms (you have seen) that a given ODE possesses
  • Make and memorize a list for yourself of the “tests” you can perform (e.g. for exact equations, including checking for integrating factors dependent on a single variable)
  • Be prepared for an exam question that lists a bunch of ODEs and asks you to write next to each ODE all the forms that it has (excluding linear subs and swapping roles, which don’t have a precise “test”)
• Families of integral curves:
  • be able to translate between an ODE and corresponding family of curves and vice versa
  • be able to find the orthogonal family of curves by translating to ODE and taking negative reciprocal slope and translating back to a family

Packet 04

• Be able to give 1-3 sentence explanation about: the connection between “integration” and “functional viewpoint”
• What are the two “primary questions” a mathematician asks about an IVP? (Existence and uniqueness.)
• What are “pathological phenomena” that can arise in the functional viewpoint with regard to existence and uniqueness? (ANS: Blowing up – non-existence; and hopping off – non-uniqueness.)
• Memorize one example IVP with solutions that blow up in finite time, and one example IVP with a family of solutions that “can hop off a line at any time.” It is acceptable to memorize examples given in the Packet.
• Can skip “infinitely differentiable non-uniqueness” example.
• Picard-Lindelöf theorem:
  • ability to state the theorem
  • understanding of the two variants
  • ability to determine whether it applies to a specific IVP to guarantee local existence and uniqueness
• Picard iteration
  • ability to execute the procedure as in the HW problem to find the first few polynomial solutions
  • don’t have to identify and/or solve recursion relations or recognize the series

Packet 05

• Tractrix and Catenary:
  • be able to derive the final ODE from the situation, as in the examples
• Pursuit Curves
  • be able to derive the final ODE from the situation, as in the examples
• Growth and decay:
  • be able to implement the correct ODE for exponential growth, including with given withdrawal rates (either constant or exponential withdrawal)
  • don’t have to fuss with taxes, changing dollar value, or gluing two scenarios end-to-end
• Autonomous equations
  • recognize and/or define an autonomous equation
  • know all terminology: phase lines, source/sink/node, stability ( ‘stability’ for numerical methods!)
  • produce/interpret phase line diagram showing qualitative solution tendency
  • identify concavity change (inflection point) as in the Example

Packet 06

• Euler’s method
  • memorize procedure and meaning
• Heun’s method
  • memorize procedure and meaning
  • memorize / ability to reconstruct Trapezoid Rule when
• Runge-Kutta (classic 4-stage as in Packet)
  • memorize procedure
• Adams method
  • memorize definition
  • memorize / ability to reconstruct “two-step” Example (fitting a parabola)
• Errors
  • terminology (local / global truncation error, rounding error, total accumulated error)
  • derivation of local truncation error for Euler’s method
• Other concepts to know the ideas for (as in the Packet)
  • adaptive methods
  • stable / unstable IVP
  • stiff IVP
  • “Q: Which solver method works for stiff IVP?” “A: Backwards differentiation methods.”

Packet 07

Second order linear, constant coefficient

• Homogeneous case: derivation of solutions
• Linearity and superposition of solutions
• Handling repeated roots using (can skip general derivation)
• Handling complex roots: exponential and trig formulations; ability to produce real solutions from complex ones

Packet 09

• Inhomogeneous theory: knowledge how to build complete family (homogeneous family + bona fide particular inhomogeneous solution)
• Variation of parameters:
  • Ability to use the method to derive a second solution for repeated roots
  • Memorize the formulas produced by the method for determining a particular solution when the homogeneous family is known
  • The complete abstract derivation is not required in either case
  • Can skip “nontrivial solution from trivial guess”
• Harmonic oscillators
  • Undamped case: memory of the polar format of solutions (amplitude and phase)
  • Overdamped case : memory of solution formula with initial
  • Underdamped case : memory of solution formula with initial
  • Can skip critical damping case
  • Driven oscillator: memory of amplitude formula; complete family; resonance phenomenon
  • ZSR and ZIR: ability to write definitions, and to solve for these formats given a complete family of solutions (for any linear ODEs, even with nonperiodic driving function)
  • Can skip driven + damped case
  • RLC circuits: can skip these facts

Packet 10

Power series

• Basic techniques for manipulating power series
• Can skip radius of convergence and root test
• Can skip example of smooth, non-analytic function
• Need ability to compute product of two series
• Know basic series list ‘Familiar examples’ but can skip and
• Be able to compute series solutions by providing a recursion formula; be very clear about small cases too
• Can skip interpreting solution in terms of familiar series, but please practice simplifying expressions for factorials and similar
• Don’t get stuck or confused on examples with strange behavior: terminating series, one free coefficient, missing negative powers (again, skipping non-analytic functions)
• Can skip example of Legendre equation etc.
• Must know: concept of ordinary point, and theorem (end of packet) that power series solutions exist near ordinary points

Packet 11

Packet 12

Linear systems basics

• Ability to convert a higher order linear equation into a linear system
• Understanding of the linear theory: homogeneous family + bona fide inhomogeneous particular solution
• Concepts of fundamental matrix of solutions, independent solutions (for case of 2 vector solutions)
• Ability to calculate the Wronskian (for case of 2 vector solutions)

Packet 13

Linear systems more advanced

• Everything for D case only on the final exam!
• Basic ability to execute the eigenvector approach for D vector solutions
• Ability to handle complex roots pair by finding the real roots
• Ability to handle insufficient eigenvalues by finding a generalized eigenvector and using the formula
• Matrix exponential technique: ability to find fundamental set of solutions by applying series to generalized eigenvectors (still D case)

Packet 14

• Fourier series
  • Memory of general form of a Fourier series
  • Even and odd functions theory; even and odd periodic extensions
  • Can skip Gibb’s phenomenon
  • Inner product of functions on an interval
  • Understanding of orthogonality; ability to derive orthogonality using the cosine sum identity
  • Basic ability to compute Fourier series of standard period functions that are in the packet: odd square wave, odd sawtooth wave, even triangle wave
• Boundary value problems
  • Good understanding of the examples with sinusoids
  • Basic eigenfunctions / eigenvalues as in the exercise
• PDE and wave equation

Excluded Problems List

All Questions / Exercises / Problems in the Packets are fair game, except those on the following lists:

Excluded Questions

• 02-02, 05-02, 05-03, 05-04, 06-03, 09-01, 09-02, 13-01

Excluded Exercises

• 02A-03
• 03A-01(b), 03A-02, 03A-04, 03B-03, 03B-04
• 04A-01 (learn the prior example instead)
• 05A-01 (learn the prior examples instead)
• 06A-01, 06B-01, 06B-02
• 07B-02, 07B-03 (learn method to produce real and imaginary solutions in specific cases)
• 09A-02
• 10-02
• 14-02, 14-03

Excluded Problems

• 02-02, 03-04, 04-03, 05-02, 06-02, 06-03
• 07-03, 07-04, 09-02, 09-04, 09-05, 09-06, 10-04, 10-05, 12-03
• 12-04, 12-06, 13-01, 13-03, 13-04, 14-03, 14-05, 14-06