# An Ode to ODEs

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## Summary

This video leads students through modeling the regular, non-linear pendulum with a differential equation. We explore why solutions to a differential equation are an infinite family of functions. We show that to determine a specific solution to this second order differential equation, two initial conditions must be specified. Proof of the need for two initial conditions is shown via use of the Taylor Series.

## Learning Objectives

After watching this video students will be able to:

• Understand that the physical laws governing a system’s properties can be modeled using differential equations.
• Explain that the solution to a differential equation is a family of functions.
• Recognize that specifying initial conditions determines a particular solution function to a differential equation.

Funding provided by the Singapore University of Technology and Design (SUTD)

Developed by the Teaching and Learning Laboratory (TLL) at MIT for SUTD

## Related Resources

Instructor Guide

An Ode to ODEs Instructor Guide (PDF)

It is highly recommended that the video is paused when prompted so that students are able to attempt the activities on their own and then check their solutions against the video.

During the video, students will:

• Describe the important forces acting on a swinging pendulum.
• Discuss whether or not differential equations have unique solutions.
• Predict whether two pendulums swinging from the same initial position will have the same behavior.
• Determine how many initial conditions are required to specify a solution for a 3rd order differential equation.