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# Differential Equations I

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

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.

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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?

2. #### Our First Equation

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

2. 2

### First-Order Separable Equations

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

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#### Modeling: The Drag Equation

Make a differential equation through experimentation.

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#### Separate and Integrate

Conquer first-order separable equations with a single technique.

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#### Application: Molecular Motor

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

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#### The Phase Portrait

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

3. 3

### Advanced First-Order Equations

Learn the power of combining geometry and differential equations

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#### Direction Fields

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

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#### Integrating Factor

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

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#### Application: A Mixing Problem

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

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#### The Potential

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

4. 4

### Basics of Linear Systems

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

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#### The Phase Plane

Learn to visualize the essential features of equation pairs.

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#### The Matrix Exponential

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

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#### Application: Underdamped Springs

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

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#### Non-Diagonalizable Matrices

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

5. 5

### Higher-Order Equations

The workhorse of physics, engineering, and applied mathematics.

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#### Equations of Order Two

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

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#### Application: RLC Filter

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

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#### Challenge: Higher-Order Equations

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

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#### Application: Hangin' Around

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