Here is a breakdown of Math 216 Exam 3 by topic. Each problem reference is a link, so you can click on it to see the problem. Your browser should take you to the right page of the exam, but keep in mind that the problem you're looking for might be toward the bottom of the page. Next to each problem you'll see a # character. If you click on that, it should take you to the solution.
Old exam problems should not be the only thing you study. They are designed to test knowledge, not teach it. And there's a danger that you might start thinking you know exactly what will be on the test. While there are some predictable topics, there will undoubtedly be questions on this year's exam which are different from any that have appeared before. So keep in mind that you're training yourself to approach hard problems, not memorizing how to do particular types.
Here's my suggestion for how to use this page:
Don't omit step 2.
I'm not entirely happy with the topics shown below. Please make suggestions on how to better categorize problems.
Let me know any errors you find, and any other feedback you have on how well it works and how to make it more useful. Links to all the exams are at the bottom of the page.
| Topic | 2001–2012 | 2013–present |
|---|---|---|
| problem involving linear systems of 3 or more equations | [2#] [5#] [1a#] | |
| problem involving phase portraits for systems of 3 or more equations | [2#] [1b#] | |
| problem involving nonlinear systems | [van der Pohl oscillator(7)#] [population(predator/prey)(8)#] [Interacting population(3)#] [5#] [8c#] [8d#] [8#] [the Brusselator(9)#] [7#] [6#] [4#] [neuron model(5a)#] [6a#] | |
| problem involving the theory of nonlinear autonomous systems | [van der Pohl oscillator(7)#] [8#] [the Brusselator(9)#] [6#] | |
| problem involving drawing or using a phase plane, including to analyze a nonlinear system | [Animal Population(6a)#] [van der Pohl oscillator(7)#] [population(predator/prey)(8)#] [Lorenz system (lab)(6)#] [Mass spring system (nl)(7)#] [2b#] [2c#] [3#] [Another van der Pol system problem(7d)#] [8#] [the Brusselator(9)#] [6#] [7#] [7#] [4#] [4#] [1#] [6#] [Engine oil(5)#] [6#] [1#] [1#] [3#] [4#] [4#] [5#] [5c#] [7#] [7#] [4#] [6#] | |
| problem involving linearization of a nonlinear system to analyze a nonlinear system | [SI (Susceptible, Infected) model(2)#] | [Interacting population(3)#] [5#] [6c#] [Lorenz system (lab)(6)#] [Mass spring system (nl)(7)#] [8#] [8#] [the Brusselator(9)#] [7#] [6#] [4#] [neuron model(5b)#] [6b#] |
| problem using a nonlinear system as a model (e.g., a population model with multiple species) | [population(predator/prey)(8)#] [Lorenz system (lab)(6)#] [Mass spring system (nl)(7)#] [7#] [4#] | |
| a problem consisting of true/false parts | [8#] [5#] | [6#] [5#] [5#] [6#] [6#] [5#] [8#] [3a#] [3b#] [3#] [8#] [8#] |
| problem involving qualitative methods for 1st order equations (direction fields, phase lines for autonomous equations) | [4#] [6#] [8#] | [5c#] [fish population with harvesting(4a)#] [Population Model(4b)#] [Population Model(4c)#] [6d#] [van der Pol equation(7d)#] [1b#] [2#] [3#] [3a#] [Engine oil(5)#] [6#] [7a#] [Chemical reaction(6)#] [1#] [5#] |
| problem involving separable 1st order equations and their solution | [2a#] [Whiffle Ball(3)#] | [1b#] [1b#] [zero entry pool(3b)#] [1#] [1c#] [4b#] [1a#] [4a#] [6a#] [1b#] [6#] [1b#] [4a#] |
| problem involving linear 1st order equations and their solution with an integrating factor | [1a#] [Whiffle Ball(3)#] | [1a#] [1a#] [1a#] [1b#] [Pollutant in Lake Michigan(3b)#] [4a#] [1b#] [Great Lakes system(2b)#] [6b#] [1a#] [1a#] [2b#] |
| problem involving the theory of solution of 1st order equations: existence and uniqueness of solutions, for linear or nonlinear equations | [4#] | [Animal Population(6b)#] [fish population with harvesting(4b)#] [fish population with harvesting(4c)#] [Pollutant in Lake Michigan(3c)#] [6a#] [6#] [5#] [2#] [3b#] [Storage tank with evaporation(3c)#] [4b#] |
| problem involving modeling with 1st order equations (e.g., population modeling, heating/cooling, etc.) | [Deer Population(7)#] | [Catapulting a Bowling Ball(4)#] [zero entry pool(3)#] [fish population with harvesting(4)#] [Pollutant in Lake Michigan(3a)#] [Population Model(4a)#] [A draining tank and a particulate solution(5)#] [Great Lakes system(2a)#] [1#] [1a#] [Engine oil(5)#] [Storage tank with evaporation(3a)#] [Storage tank with evaporation(3b)#] [2a#] |
| problem involving linear systems of algebraic equations (e.g., x + 2y = 3, 3x - y = 5) | [6b#] [2a#] [3#] [4c#] [4#] [7c#] | |
| problem involving linear systems of differential equations (e.g., x' = a(t) x + b(t) y, y' = c(t) x + d(t) y, or x' = x + 2y, y' = 3x - y) | [2#] [van der Pol equation(7a)#] [1#] [2b#] [2c#] [3#] [Great Lakes system(2c)#] [6#] [7#] [4b#] [4d#] [4b#] [4c#] [5a#] [5b#] [7a#] [7b#] [3#] | |
| problem using the eigenvalue method to solve a linear system, where eigenvalues are real | [2a#] [6#] [2a#] [5a#] [5b#] [van der Pol equation(7c)#] [1a#] [4odelinsystemsa#] [7#] [7a#] [2a#] [Engine oil(5)#] [2a#] [7#] | |
| problem using the eigenvalue method to solve a linear system, where eigenvalues are complex | [2b#] [2b#] [5c#] [4b#] [6#] [7b#] [2b#] [4c#] [7b#] [2b#] [4#] [6b#] | |
| problem using the eigenvalue method to solve a linear system, where eigenvalues are real and repeated | [1b#] [5#] [7#] [6a#] | |
| problem using the eigenvalue method to solve a linear system; use if one of the previous three isn't obviously appropriate | [2#] [5#] [6c#] [2#] [Another van der Pol system problem(7c)#] [Another van der Pol system problem(7d)#] [Great Lakes system(2)#] | |
| problem involving the theory of solutions of systems of linear equations | [4a#] [4b#] [2#] | |
| problem involving nonhomogeneous linear systems (e.g., x' = x + 2y + f(t), y' = 3x - y) | [7#] [5#] [1#] [7c#] [1#] [7#] | |
| problem involving linear second order, homogeneous equations | [7#] | [Mass spring system(4)#] [1#] [3#] [5#] [4#] [6#] [3a#] [3b#] [4a#] [4b#] [4c#] [6#] [1#] [4a#] [6#] [1#] [3#] [5a#] [5b#] [7a#] [7b#] [7d#] |
| problem involving the theory of second (or higher) order linear equations | [5a#] [Linearity redux(2)#] [3#] [5#] [6#] [3#] [3#] [3#] [4#] [4#] [7a#] [3a#] [3b#] [5#] [5c#] [7#] [3#] | |
| problem involving modeling with 2nd (or higher) order linear equations without forcing (e.g., spring problems without forcing) | [Mass spring system(4a)#] [Mass spring system(4b)#] | [Mass spring system(4)#] [More Lasers(7)#] [4#] [4#] [4#] [4#] [6a#] [2#] |
| problem involving modeling with 2nd (or higher) order linear equations (e.g., spring problems with forcing) | [7#] | [Chemical Reaction(3)#] [RLC Circuit(4)#] [Laser Model(7)#] [5c#] [6#] [3b#] [Another ruby laser(4)#] [Another ruby laser(4c)#] [6b#] [More Lasers(7c)#] [5#] [4b#] [4c#] [3#] [1a#] [1b#] |
| problem involving the use of the method of undetermined coefficients | [1a#] [1#] [1#] [1a#] [1b#] [1b#] [3a#] [5a#] [4#] [6#] [1#] [7b#] [1#] [3#] [1#] [4#] | |
| problem involving the use of variation of parameters | [1b#] [6a#] [Another ruby laser(4b)#] [3c#] [3d#] [5#] | |
| problem involving the definition of the Laplace transform | [3b#] [6b#] [2#] [1a#] [2c#] [4a#] [3#] | |
| problem involving the linearity or other properties of the Laplace transform | [5b#] [5b#] [5d#] [5e#] [6#] [7c#] [2#] [3a#] [1b#] [2#] [5#] [4b#] [6#] [7#] | |
| problem using laplace transforms to solve a differential equation | [2a#] [2b#] [2a#] [2b#] [3a#] [5b#] [6c#] [7#] [1#] [3b#] [2a#] [2b#] [7#] [2#] [5#] [7c#] [2#] [4c#] [6b#] [6c#] [7c#] [6#] [7#] | |
| problem using laplace transforms to solve a differential equation with discontinuous (step function) or impulse (delta function) forcing | [5c#] [6b#] [2c#] |
All Exam 3's, with their medians:
| Fall, 2012 | Exam | Solutions | 71 | Winter, 2013 | Exam | Solutions | 68 |
| Fall, 2016 | Exam | Solutions | 63 | ||||
| Winter, 2017 | Exam | Solutions | 75 | ||||
| Fall, 2017 | Exam | Solutions | 71 | ||||
| Winter, 2018 | Exam | Solutions | 52 | ||||
| Fall, 2018 | Exam | Solutions | 72 | ||||
| Winter, 2019 | Exam | Solutions | 70 | ||||
| Fall, 2019 | Exam | Solutions | 68 | ||||