MATH 224 - CHECKLIST FOR FINAL EXAM (Tuesday, December 8, 4 - 7 p.m. in KHSM 123, the usual classroom)

• recognize standard classes of first order equations (separable, linear, etc.) and solve them, if feasible
• solve homogeneous linear systems in two dimensions (eigenvalues/eigenvectors etc.)
  In both cases "solve" means either "give a general solution" or "solve an initial value problem."
• calculate Laplace transforms or inverse Laplace transforms, or to use them to solve an initial value problem
• perform a qualitative analysis of an equation or of a system
• a modeling problem (set up an equation/initial value problem and solve it)

• Chapter 1 except for sections 1.4 and 1.7
• Chapter 2, sections 1-4 and 6
• Chapter 3 except for section 3.7
• Chapter 4, sections 1-3
• Chapter 5, sections 1-3
• Chapter 6, sections 1-5

• Modeling (a word problem): population models, fish ponds, mixing problems...
  Sample exercises: 1.1:17,18; 1.2:39,42; 1.9:24-27; 2.1:1-6; Review Ch.1:53, 54
  
  
• Recognizing known classes of first order equations (separable, linear, etc.) and solving equations/initial value problems analytically, if feasible
  Sample exercises: Review Ch.1:21-39
  
  
• Linear systems/equations (2.3, 2.4; chapter 3):
  
  Primary skill :
  • finding general solutions and solving initial value problems for homogeneous linear systems in two dimensions
  Exercises for this: solve one or more problems for each pattern of eigenvalues: two real, one (repeated) real, two complex. It goes without saying that you must understand well what eigenvalues/eigenvectors are.

Auxiliary skills/concepts :

• converting second (or higher) order equations to a system of first order equations
• general understanding of linear systems in three dimensions
• solving analytically decoupled and partially decoupled systems
Note that while we did not cover in detail section 3.7, you still need to be able to classify equilibrium points; this will be particularly important in qualitative analysis of systems (linear or nonlinear).
Sample exercises: 2.4:13; 3.8:17; Review Ch.2:3; Review Ch.3:5,8,9,23-26


• Second order equations with or without forcing terms: you will be asked either to solve an initial value problem or to give a general solution. There may be some physical jargon (harmonic oscillators, resonance, beating, beating frequency) that needs to be recognized and understood if it occurs. Besides the guessing method, Laplace transform may also be of use.
  Sample exercises: 3.6:13-18; Review Ch.4:10, 15-23
  
  
• Qualitative analysis (1.3, 1.6, 2.2, 5.1, 5.2, and also some aspects of sections 3.3, 3.4, 3.5; mostly, but not exclusively, for autonomous equations). Some of the items below apply to scalar equations, some to systems and some to both:
  • sketching phase lines; rough sketches of solution graphs based on phase lines or slope fields
    Sample exercises: 1.6:1-12; 1.3:7-10
  • matching equations with slope fields/phase lines
    Sample exercises: 1.3:16; 1.6:37
  • finding equilibria and classifying them for scalar equations and for linear or nonlinear systems
    Sample exercises: Review Ch.2:2,5,35; 3.3:1-12; 3.4:3-8; Review Ch.3:6; 5.1:1,3,11; Review Ch.5:1,2,15a,16a,26
  • qualitative analysis of long term behavior of solutions for scalar equations and for linear or nonlinear systems; this may involve nullclines etc.
    Sample exercises: 1.6:13-27; 5.2: 1,7,9; Review Ch.5:5,15-18
  • understanding phase planes; matching phase planes, x(t)/y(t)-graphs and systems
    Sample exercises: 2.2:11, 21; Review Ch.2:31-34; Review Ch.3:19,21
  
  
• Laplace transform and applications: you must know how to find both the Laplace transform of a function and the inverse Laplace transform of a function. [This may be needed to solve a differential equation, or asked as a separate problem.] The transforms are either to be found from the definition or (more typically) extracted from the "dictionary" of standard Laplace transforms using the learned rules and tricks such as
  • linearity of the transforms
    
  • the effects of translations and multiplication by an exponential function
    
  • the effects of differentiation (in either variable, t or s)
    
  • expressing functions defined "interval-wise" by a single formula via step functions
    
  • and, yes, partial fractions, "delta functions" etc.
    Sample exercises: Review Ch.6:3, 5, 6, 11, 18-20, 25, 27, 30
  
  
• Hamiltonian systems: you may be asked to verify whether a system is or is not Hamiltonian and to find its Hamiltonian function. It is also possible that you will be asked to perform (elements of) qualitative/equilibrium analysis for such systems, which may include coming up with a rough sketch of the phase space.
  Sample exercises: 5.3: 3, 10-13; Review Ch.4:6, 25a-c
  
  
• It goes without saying that you need to understand the basics of differential equations:
  • when a given function (scalar or vector) is a solution
  • when a given graph represents a solution
  • same for initial value problems
  • checking the hypotheses of, and drawing conclusions from the existence/uniqueness theorem