## Math 285 - Final Exam Information

#### (last updated May 5)

Exam Details:
• usual classroom
• Friday May 7 from 1:30 to 4:30 PM
• comprehensive
Checklist for the Exam:
Concepts and Theory
You should be familiar with the following theoretical results:
• interpretation of a 1st order equation as a formula for the slope of solutions
• definition of exactness of M + Ny' = 0, test for exactness
• statement of the main existence and uniqueness theorem for higher order linear differential equations 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 and relationship with genuine constants
• statement of Theorem III on the Wronskian test for linear independence
• 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?)
• know the relationship between even and odd functions and sine and cosine series, as well as even and odd extensions of functions
• 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
• 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 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
• 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
• Applications
You should be able to solve the differential equations and interprete results in connection with the applications:

• Newton's law of cooling
• resistance problems and terminal velocities
• mechanical vibrations of a simple mass-spring-resistance system. In particular this means being familiar with associated concepts such as
• natural frequency of oscillation
• frequency, period, etc
• underdamping, overdamping, critical damping and how they arise
• transient part and steady state solution for a forced oscillator
• solving boundary value problems for the heat, wave, and Laplace's equation, including setting up such problems so as to handle insulated boundaries, etc.