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Ordinary Differential Equations MA26600 Spring 2026

Jakayla Robbins

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Worksheet Lesson 1, Introduction to Differential Equations

Textbook Section(s).

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This lesson is based on Section 1.1 of your textbook by Edwards, Penney, and Calvis.
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Introduction to the Course.

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You should be able to access everything you need for the course through Brightspace. If you find that something is missing from Brightspace, please let me know. In particular, you can find information about the following:
  • course syllabus, including:
    • A link to the departmental course web page that contains:
      • the Ground Rules for the course (grading scale, homework policies, calculator policies, etc.)
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      • a list of assignments
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      • basic exam seating charts (I will give you specific seat assignments in the Brightspace Gradebook later in the course.)
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      • some resources (MyLabMath Quick Student Guide, computer programs that will be useful throughout the semester, past exams, practice problems for exams, etc.)
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    • course calendar, including:
      • test dates and times (The date and time of the final exam are not yet available, but I will post them in Brightspace as soon as they are available.)
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      • due dates for online homework assignments
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      • due dates for written assignments
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    • information about Dr. Robbins’ office hours
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    • information that should be included in any email that you send to Dr. Robbins.
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  • links for MyLab Math where you will complete online homework assignments
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  • links for Gradescope where you will submit written homework assignments
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  • links for the sections of the e-book that we will cover this semester
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  • A page dedicated to lecture materials, including:
    • notes templates for any lectures that will utilize notes templates
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    • completed lectures, when available
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    • any other links that I think are relevant to a lecture
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  • a few extra resources that you might find useful during the semester
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  • any announcements for the class
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What is a differential equation?

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Definition 1.

A differential equation (DE) is an equation that contains at least one derivative. If \(\frac{d^ny}{dx^n}\) (or \(y^{(n)}\)) is the highest order derivative in the differential equation, then the differential equation is called an nth-order differential equation.
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Example 2. Terminology about differential equations.

All of the following are differential equations.
  1. \(\displaystyle \frac{dy}{dx}=x^2+1\)
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  2. \(\displaystyle \frac{dy}{dx}=x^2y^2\)
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  3. \(\displaystyle \frac{d^2y}{dx^2}+\frac{dy}{dx}+3=0\)
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  4. \(\displaystyle \frac{\del f}{\del x\del y}- x\frac{\del f}{\del x}=x^5\)
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Equations are first-order differential equations.
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Equations (c) and (d) are .
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Equation (d) is a partial differential equation.
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Definition 3.

A solution of a differential equation is a function that satisfies the differential equation.
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Example 4. Solving a simple differential equation.

Find a solution of the differential equation.
\begin{equation*} \frac{dy}{dx}=x^2+1 \end{equation*}
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Example 5. Verifying a solution of a differential equation.

Show that \(y=\frac{-3}{x^3+C}\) is a solution of the differential equation \(\frac{dy}{dx}=x^2y^2\text{.}\)
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Initial Value Problems.

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When you are asked to solve a differential equation, the solution is really a of solutions, as indicated by the undetermined constant(s) in the solution. We call these families of solutions . If we want a specific solution, then we need more information. In particular, we need some specific input and output combination(s) for the function and/or its derivatives. These specific input/output combinations are called the . We will use the initial values to find specific values for the constant(s) in the general solution.
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Example 6. Finding the constant in and IVP.

Given that \(y=\frac{-3}{x^3+C}\) is the general solution of the differential equation \(\frac{dy}{dx}=x^2y^2\text{,}\) solve the intitial value problem.
\begin{equation*} \frac{dy}{dx}=x^2y^2, \qquad y(0)=1\text{.} \end{equation*}
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Modeling with Differential Equations.

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Derivatives are formulas for rates of change, so they appear in equations for lots of applications.
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Example 7. Modeling a rate of change.

The rate of change of a population is proportional to the size of the population.
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Example 8. Using tangent line information in a model.

(Number 28 from Section 1.1 of your textbook by Edwards, et.al.) The line tangent to the graph of \(g\) at the point \((x,y)\) intersects the \(x\)>-axis at the point \(\left(\frac{x}{2},0\right)\text{.}\) Write a differential equation of the form \(\frac{dy}{dx}=f(x,y)\) for which \(g\) is a solution.
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