# Differential Equations I

## The math of change, from economics to physics.

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#### Overview

Differential equations show up in just about every branch of science, including classical mechanics, electromagnetism, circuit design, chemistry, biology, economics, and medicine.

From analyzing the simple harmonic motion of a spring to looking at the population growth of a species, differential equations come in a rich variety of different flavors and complexities.

### Topics covered

• Applications in Engineering
• Applications in Physics
• Direction Fields
• Euler's Method
• Integrating Factors
• Linear Systems
• Matrix Exponential
• Modeling
• Perturbation Method
• Phase Portraits
• Separable First-Order Equations
• Wronskian Determinants

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1. 1

## Introduction

Discover why differential equations are so important!

1. ## Differential Equations in a Nutshell

What are differential equations and what are they used for?

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2. ## Our First Equation

Use Newton's Law of Cooling to build and solve your first differential equation.

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3. ## A Quick Review

Review the algebra and calculus tools you'll need for the road ahead.

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2. 2

## First-Order Separable Equations

Build and explore real-world models with first-order equations.

1. ## Modeling: The Drag Equation

Make a differential equation through experimentation.

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2. ## Separate and Integrate

Conquer first-order separable equations with a single technique.

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3. ## Application: Molecular Motor

Unravel secrets of the biological micro-world with first-order equations.

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4. ## The Phase Portrait

Learn to sketch integral curves like a pro by solving this chemistry challenge.

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6. ## Application: In the Chem Lab

Practice your first-order skills while exploring this chemistry application.

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7. ## Capstone: Vampires of Cancelvania

Use differential equations to save the countryside from a vampire plague!

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3. 3

Learn the power of combining geometry and differential equations

1. ## Direction Fields

Use a simple, geometric technique to draw integral curves on the phase plane.

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2. ## Integrating Factor

Combine vectors and geometry into a technique for solving first-order problems.

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3. ## Application: A Mixing Problem

Apply the integrating factor to a classic real-world problem.

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4. ## The Potential

Explore the meaning of the integrating factor in the context of physics.

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5. ## Application: Fluid Flow

Explore the potential for using first-order equations to make real-world models.

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6. ## Capstone: The Great Escape

Drop in on a marooned astronaut who uses first-order equations to call for rescue.

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4. 4

## Basics of Linear Systems

Matrices and vectors unlock the gateway to higher-order equations.

1. ## The Phase Plane

Learn to visualize the essential features of equation pairs.

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2. ## The Matrix Exponential

Learn how to use vector geometry as a powerful tool for exploring linear systems.

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3. ## Application: Underdamped Springs

Gain hands-on experience with the matrix exponential in a familiar setting.

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4. ## Non-Diagonalizable Matrices

What do you do when you just don't have enough eigenvectors?

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5. ## Review: Math of a Salesman

Practice your linear systems skills on a mind-altering robot sociology problem.

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6. ## Nonhomogeneous Systems: Part I

Make an atom smasher by combining physics with systems of equations.

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7. ## Nonhomogeneous Systems: Part II

Make a master formula for linear systems and complete your cyclotron.

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8. ## Challenge: Floquet Theory

Strengthen your grasp on linear systems while learning how medical NMRI works.

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5. 5

## Higher-Order Equations

The workhorse of physics, engineering, and applied mathematics.

1. ## Equations of Order Two

Uncover the deep connection between higher-order equations and systems.

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2. ## Application: RLC Filter

Find out what second-order equations can do for electrical engineers.

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3. ## Challenge: Higher-Order Equations

Reach beyond second-order and uncover a master formula with the Wronskian matrix.

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4. ## Application: Hangin' Around

Solve a structural engineering problem with a third-order equation.

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5. ## Application: Beam Me Up!

Can a building withstand an earthquake? Find out with fourth-order equations!

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6. ## Application: Get Your Motor Runnin'

Retrieve a treasure from the briny depths with a third-order equation.

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7. ## Challenge: Why's The Sky Blue?

Answer a very common science question by applying the perturbation method.

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