## Study Guide for Final Exam

**Administrative Details**
The exam is Tuesday, 10 Aug from 10:30am-12:30pm.
You will be able to enter the Zoom room (same as class) at least 15 minutes before start time. Please use your first and last name when you connect and activate your video showing the testing configuration described below.
You will use a two-device proctoring system as mentioned in the syllabus and outlined in This document. The following will occur 5 minutes before the exam starts (all on my cue):
- Show your face for ID checking
- Scan your work area to show everything is clear
- Show your cover sheet/Laplace Transform table (If you print it out: front and back)
- Show all other paper is blank
- Place your camera/phone so I can see your computer screen and your work area. Hands must be in view at all times.

The exam will consist of 8 workout problems worth 8-16 points each. Some problems will have
multiple parts. All answers must be justified with appropriate work and/or explanation. Partial credit will be given for appropriate work shown on these problems.

Correct answers without sufficient work will receive ZERO credit and may result in filing an Honor Code violation.
**IMPORTANT!!!**: If you are using your own paper, make sure it follows the Template linked here (UPDATED 6 Aug!!!), including Name and UIN on the first page. The letters do not have to be exactly placed, but should be approximately located.
Calculators are not allowed, BUT I will provide you with any complicated formulas you may need for the exam (as well as a copy of the entire Laplace Transform table posted in Canvas). I expect you to know all precalculus material (including the unit circle), all derivative rules, and all integration rules except non-tabular Integration by Parts, Trig Integrals (other than sine, cosine, and simple substitutions), and Trig Substitution.
If you have questions or issues that come up during the exam, you may send me questions in the Zoom chat (preferable) or email me. I will answer what I can.
Once you have finished your exam, let me know in the Zoom chat. When I respond, you will scan your work to a PDF as per usual and upload the PDF to Gradescope.
**IMPORTANT**: If you print the exam and write on it, do NOT print the cover sheet! You should submit exactly 8 pages to Gradescope, or it will not accept your submission. If you need additional space, continue on one of the other pages (clearly indicating which in your work).
I recognize that you may have to shut off your video during this time. You should be able to scan and upload in about 10 minutes. If you are experiencing technical issues, let me know (Zoom chat or email) so I know that is the cause of the delay.
During the exam you should only have access to: Writing Utensils, Picture ID, blank/templated paper. All other materials should be out of sight during the exam.

**Learning Outcomes** (from Syllabus. Outcomes NOT tested on the final exam are struckthrough):
Students will use differential equations to model mechanical and electrial systems.
~~Students will visualize solutions to first order differential equations and 2 x 2 systems of first order linear differential equations using direction fields and phase planes~~
Students will solve basic first order differential equations and initial-value problems
Students will understand the conditions required for a first order differential equation to have a unique solution
~~Students will find the equilibrium points of an autonomous differential equation and determine their stability~~
Students will solve homogeneous second-order linear differential equations and initial value problems with constant coefficients.
Students will use the Method of Undetermined Coefficients and Variation of Parameters to find solutions to nonhomogeneous second order linear differential equations and initial value problems with constant coefficients.
Students will use Laplace Transforms to solve basic initial value problems
Students will determine the mathematical and practical effect of step functions and impluse functions on second order linear initial value problems with constant coefficients.
Students will use Power Series to solve second order linear differential equations.
Students will write a higher order differential equation as a system of first order differential equations
Students will solve homogeneous systems of first order linear differential equations
Students will conduct qualitative analysis of 2 x 2 systems of linear first order differential equations with constant coefficients
~~Students will understand methods of numerically approximating solutions to first order initial value problems~~

**Topics by Section** (sections NOT tested on the final exam are struckthrough):
~~1.1-1.3 Modelling, Direction Fields~~
2.1 Linear First Order ODEs
2.2 Separable ODEs
2.3 Applications of First Order ODEs
~~2.4 Existence and Uniqueness of Solutions~~
~~2.5 Autonomous ODEs~~
2.6 Exact Equations
3.1 Homogenous Second Order ODEs with Constant Coefficients
3.2 The Wronskian
3.3 Complex Roots of the Characteristic Equation
3.4 Repeated Roots of the Characteristic Equation
3.5 Method of Undetermined Coefficients

3.6 Variation of Parameters
~~3.7 Applications of Second Order ODEs~~
~~3.8 External Forcing Functions~~
~~6.1 Definition of Laplace Transform~~
6.2 Solving IVPs with Laplace Transforms
6.3 Heaviside Functions and Laplace Transforms
6.4 Solving ODEs with Heaviside Forcing Functions
6.5 Solving ODEs with Impulse Forcing Functions
6.6 Convolution Integrals
5.2 Series Solutions, part 1
5.3 Series Solutions, part 2

7.1 Introduction to First Order Linear Systems
7.2 Introduction to Matrices, part 1
7.3 Introduction to Matrices, part 2
7.4 Existence and Uniqueness of Solutions to First Order System IVPs
7.5 Solving ODE Systems (distinct real eigenvalues)
7.6 Solving ODE Systems (complex eigenvalues)
7.8 Solving ODE Systems (repeated real eigenvalues)

**Approximate Breakdown of Material**
Exam I Material 35%
Exam II Material 35%
Post-Exam II Material (ch 7) 30%

**Suggestions for Studying**:
**For past material: I recommend using your old exams as a starting point. If you were unable to solve a problem then, or don't recognize how you solved it now, go back and review online/suggested HW over that topic. For new material (ch 7), online and suggested HW are a good source of preparation. **
In addition:
Read your textbook, including all Examples
Read your notes from lecture, including all Examples
Suggested Homework Problems
Online Homework (Practice versions of all past due HW sections available)
Week in Review (Archived from Spring 2021)
Help Sessions end Mon, 9 Aug. (Links available at the Math Learning Center page linked here)
Regularly scheduled Office Hours end Mon, 9 Aug.

I will also hold hours Mon, 9 Aug 2-4pm OBA