Exam Details:

• usual classroom
• Monday, December 11, 1:30pm-4:30pm
• comprehensive (will cover ALL the material of this course!)

An old version of a final exam for this course
 

Allowed materials:

•  TWO   8''x11''  sheets of notes and formulas, each of which may be two-sided
• calculator

Checklist for the Exam:

Concepts and Theory

You should be familiar with all those statements which are presented as "Theorems" in the book. In particular, pay attention to the following:

• definition of exactness of M + Ny' = 0, test for exactness
• statement of the main existence and uniqueness theorem for higher order linear differential equations (Theorem 2 on p. 156) and its significance for problems we solve (i.e. how many initial conditions should we expect to impose, what intervals I in the indep variable x ensure existence and uniqueness, etc.)
• definition of linear independence
• All theorems from Ch 3.2 and 3.3
• know the formulas for the coefficients in sine series, cosine series, and full Fourier series
• understand the convergence theorem for Fourier series (i.e.can you draw a graph of the function your series really represents?) and the eigenfunction series (Ch. 10.1)
• know the relationship between even and odd functions and sine and cosine series, as well as even and odd extensions of functions
• statements regarding termwise differentiation and integration of Fourier series
• know what is meant by an eigenvalue problem for a differential equation and what you are trying to find when you solve such a problem
• definitions of orthogonality in the context of Fourier series and Sturm-Liouville problems


Techniques
You should be proficient with calculations dealing with:

• identifying and solving 1st order separable equations (and solve for y if you can!)
• identifying and solving 1st order linear equations
• testing equations for exactness and solving equations that are exact
• solving constant coefficient higher order equations via characteristic equations, etc. (This means being able to handle distinct roots, repeated roots and complex conjugate roots.) Be sure you know where the characteristic equation comes from!
• know what differential operator notation means and be able to handle problems written in operator form
• solving non-homogeneous constant coefficient equations via the method of undetermined coefficients
• checking for linear independence using the Wronskian test. You should be able to hand both 2x2 and 3x3 determinants
• the method of variation of parameters for solving non-homogeneous problems
• computing sine, cosine, and full Fourier series
• finding a Fourier-series solution of a differential equation
• solving simple eigenvalue problems (you may be told that the eigenvalues are non-negative to simplify the calculation, so pay attention)
• applying the separation of variables method to find all "separated solutions" of a PDE with boundary conditions
• applying superposition ideas to separated solutions and using this to satisfying initial conditions via Fourier series and eigenfunction series
• you may and are expected to use the formulas for solving heat equation problems, wave equation problems and Dirichlet problems which are proved in the appropriate chapters of the book. Given such a problem you should APPLY these formulas rather than derive them from scratch. However, if you are given a problem which does not fall into one of these categories, you should provide a complete solution starting with the separation of variables (e.g. problems from the exercise set for Ch. 10.2).

 
Applications
You should be able to solve the differential equations and interprete results in connection with the applications:
 
• Newton's law of cooling and Torricelli's Law (Ch 1.4) and mixture problems (Ch. 1.5)
• mechanical vibrations of a simple mass-spring-resistance system (undamped motion only) (Ch. 3.4 and 3.6). In particular this means being familiar with associated concepts such as
• natural frequency of oscillation
• frequency, period, etc
• resonance
• solving boundary value problems for the heat, wave, and Laplace's equation, including setting up such problems so as to handle insulated boundaries, etc.

                               Here is a cpmplete list of chapters which will be covered by the Final Exam:
 

                              1.1-1.6,  3.1-3.6, 3.8, 9.1-9.7, 10.1-10.2