(MATH122 - Calculus II)
Author: Kris Hollingsworth, PhD (Dr. ℋ)

Class 02: Section (7.7)
Beginning Ordinary Differential Equations

Today’s Plan
  • Discuss Course Schedule and How You Earn Grades
  • Discuss Learning Journals.
  • Learn basic concepts and definitions of [Ordinary] Differential Equations (ODEs)
  • Work on practice problems for ODEs, time permitting.

Note: The course syllabus, daily schedule, and other course information is always available on the course website, D2L.

Today’s Assignments

Looking Ahead: Major Milestones Coming Up.
  • Foundations Assessment 01: Algebra, Trig, and a bit of Calc 1
    • Receive today.
    • Due this Friday.
  • Foundations Assessment 02: Derivatives and Integrals
    • In class second Friday of semester.
    • Calculate 10 derivatives similar to the extra practice on D2L under Guided Practice 01.
    • Calculate 10 integrals similar to Guided Practice 03 >This will be Thursday’s assignment.
  • Exam 01: Fourth Friday of Semester
    • Covers Review material and applications of integration (we’ll start that today).

Grading Scheme
Course Grading Scheme
Activity Weight Description
Attendance 2% Show up to class. On time. Attendance will be taken at the beginning of class.
Foundations Assessment 1 3% Take-home, open resource review.
Foundations Assessment 2 3% In-class assessment on Calculus I derivatives and integrals.
Group Assessments 5% In class exploratory or review assessments completed in small teams. Assessment will be discussed more on Thursday of the first week, when the first group assessment occurs.
Guided Practice 5% Short 2-4 page, low-stakes take-home assessments to get feedback on your mathematical communication and to identify potential sticking points. Generally graded generously if you make a real effort, these are your chance to make mistakes and get written feedback. Ignoring that feedback generally results in poor marks on higher-stakes assessments.
Learning Journals 8% Your primary method for practicing and developing mastery of course content. You will keep a journal of completed learning problems. You are expected to complete every odd problem and at least half the even problems to receive full credit.
Exams 60% (5%, 10%, 12%, 15%, 18%)
5 in-class chapter exams dynamically weighted in your favor. Five Problems will be released the day before, 3 of which will make up ~40-50% of the in-class exam grade. The rest will be similar to material from the course notes, guided practice, learning journals, or group assessments. Yes. It’s all important.
Final Project 14% A final summative, creative work in which you will summarize and organize your learning from the semester. Guidelines, an assessment rubric, and an initial brainstorming Guided Practice will be released a little later in the semester.
Final Exam 0% Comprehensive final exam. See details in assignment description, score on final exams can raise or lower your final course grade by up to two-shades. Even currently failing, a 85% or higher will result in at least the minimal passing course grade.

Learning Outcomes
  • Identify differential equations and their order.
  • Define general and specific solutions of differential equations.
  • Verify general solutions of DEs are solutions.
  • Find a particular solution of a differential equation given a general solution.

Differential Equation

A differential equation (DE) is an equation containing an unknown function and some of its derivatives.

Order (of a DE)

The order if a DE is the order of the highest derivative that occurs in the equation.

Find the order of each DE.

Solution (of a DE)

A solution of a DE is a function which satisfies it.

Exercise

Solve the DE y′=2xy'=2x.

Hint (and a question)

Use the fundamental theorem of calculus or your knowledge of derivatives to determine the solution. Why are there infinitely many solutions?

Exercise

Given y′=2xy'=2x, which solution corresponds to the initial conditions [ICs]:

  • y(0)=1y(0)=1
  • y(−1)=3y(-1)=3
  • y(2)=1y(2)=1

Visual Aid

Motivation

Many applications in math require DEs!

Often: We observe the way a system changes as a function of time/space, then we develop a mathematical model using DEs to describe/predict the system.

Modeling examples

Math Problem Example

Population Growth Let P(t)P(t) represent the current population of a bacteria colony at time tt. Suppose the population grows at a rate proportional to the current population.

  • Write a DE describing the system.

Answer

Let kk represent the proportion. Then the differential equation is given by dPdt=kP,\frac{dP}{dt} = kP, where PP is the current population as a function of time and k>0k>0 is a constant.

  • Verify the general solution is P(t)=AektP(t)=Ae^{kt}.

Approach

To verify the solution, always start from the given information, and use it to show the mathematical property or equation is true based on that information. In this case, we need to take the derivative: P′(t)=AkektP'(t)=Ake^{kt} and then algebraically manipulate it to look like the equation we want to verify is true based on the given information.

Caution

If you are told to verify something is a solution, that means following the specific approach given above. Solving the problem analytically will not earn any credit in this situation, as you are simply not answering the specific question that was asked.

Start on Individual Practice Problems

Time permitting, you may just be starting these on your own outside of class.

Next Class

Our first group assessment reviewing integration using uu-substitution and other basic concepts!