Exam info & Solutions
Information about Exam I — Exam I will be held in the usual room at the usual class time, on Wednesday February 14. It will start promptly at the beginning of the class period, so make sure you're on time. The test will cover sections 1.1 - 1.6. Note that in section 1.6 we did not cover the part on "Exact Differential Equations" on pages 64-67, so this will not be covered on the exam.
The test will have four or five problems, possibly with several parts each.
The following topics are likely to be covered, though the exam is not limited to these topics:
- checking whether a function is a solution to a differential equation; solving for initial conditions
- slope fields for certain first-order equations
- using separation of variables to solve equations
- first-order linear equations (via integrating factors)
- first-order homogeneous equations (via substitution)
- Bernoulli equations (via substitution)
- other substitutions
- solving real-world problems (eg. heating and cooling, mixing problems, etc) using these methods. The relevant differential equation will be given in some form.
- solving reducible second-order equations
You do not need to memorize difficult integration formulas or trig formulas. If these are needed, I will provide a table for you. You are expected to know the basics, however.
Advice: I recommend first reviewing homework problems from each section. You should have done all the assigned problems at the very least, and trying others as well is a good idea. If you get one wrong, don't skip ahead, but tackle it right away and find out how it works. There may be a similar example worked out in the book, or among one of our graded problems, or you can try getting help. There are also extra problems at the end of each chapter, under the heading "Review Problems" which are worth looking at.
When it comes time for the test itself, try to relax and do the best you can. At this point, being calm and well-rested will be important. Also, scan through the exam problems and identify the easier ones to get out of the way first.
Information about Exam II — Exam II will be held in the usual room at the usual class time, on Wednesday March 14. It will start promptly at the beginning of the class period, so make sure you're on time. The test will cover sections 3.1 - 3.5. Note that in section 3.5 we did not cover the part on "Variation of Parameters" on pages 192-194, so this will not be covered on the exam.
The test will have the same format as the first exam.
The following topics are likely to be covered, though the exam is not limited to these topics:
- the existence and uniqueness theorem for solutions to linear differential equations
- using the Wronskian to determine linear independence of solutions
- Solving constant-coefficient linear homogeneous equations using the characteristic polynomial (real roots and/or complex roots)
- Finding a particular solution to a non-homogeneous linear equation, by the method of "undetermined coefficients" (watch out for "duplication")
- Finding the general solution to a constant-coefficient non-homogeneous linear equation
- applications to mechanical vibration problems
- combining two oscillations of the same frequency into a single oscillation (with amplitude, circular frequency, and phase angle)
As before, you do not need to memorize difficult integration formulas or trig formulas. If these are needed, I will provide a table for you. You are expected to know the basics, however.
The advice for Exam I also applies here.
Information about Exam III — Exam III will be held in the usual room at the usual class time, on Wednesday April 18. It will start promptly at the beginning of the class period, so make sure you're on time. The test will cover sections 7.1 - 7.5.
The test will have the same format as the first exam.
I will provide a list of Laplace transform rules and formulas, roughly equivalent to the inside front cover of the textbook. So you do not need to memorize these. However, you will want to be sure you are comfortable applying the rules correctly.
The following topics are likely to be covered, though the exam is not limited to these topics:
- the definition and basic properties of the Laplace transform
- computing Laplace transforms, using the definition, and using various properties
- finding inverse Laplace transforms
- solving initial value prooblems using the Laplace transform
The advice for Exam I also applies here. Good luck!
Information about the Final Exam — the Final Exam will be held in the usual room from 8:00 to 10:00 am on Tuesday May 8. It will start promptly at 8:00, so make sure you are on time.The test will cover all of the topics of the course listed above, as well as the new material: sections 7.6, 4.1, 4.2, 5.1, and 5.2. In section 5.1, the subsection titled "Initial Value Problems and Elementary Row Operations" will not be covered on the exam. In section 5.2, the subsections "Compartmental Analysis" and "Complex Eigenvalues" will not be covered.
The test will have a similar format to the previous exams, but will be slightly longer. Some emphasis will be given to the new material.
I will provide a list of Laplace transform rules and formulas, from the inside cover of the book, so you do not need to memorize these. However, you will want to be sure you are comfortable applying the rules correctly. The list looks like this.
From the new material, the following topics are likely to be covered:
- Laplace transforms of impulses, and applications to initial value problems
- first-order systems of equations, such as multiple mass-spring systems
- turning a higher-order equation into a first-order system
- reducing a system to a single equation using elimination
- expressing linear systems using matrices and vectors
- linear independence of solutions to systems, and using the Wronskian (note, the Wronskian has a new definition in this setting)
- finding eigenvalues and eigenvectors of matrices
- using eigenvalues to solve homogeneous linear first-order systems
Office hours: Monday May 7, 9:30 - 11:30 am.