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# Differential Equations II

## The frontier of scientific modeling.

Drum vibrations, heat flow, the quantum nature of matter, and the dynamics of competing species are just a few real-world examples involving advanced differential equations.

These models and many others from across the sciences, engineering, and finance have nonlinear terms or several independent variables. Their equations hold many surprises, and their solutions draw on other areas of math like linear algebra and vector calculus.

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1. 1

### Introduction

What are nonlinear and partial differential equations?

1. #### Nonlinear Equations in a Nutshell

What makes an equation nonlinear, and why does it matter?

2. #### PDEs in a Nutshell

Tour the equations governing heat, waves, diffusion, and the quantum realm.

2. 2

### Nonlinear Equations

The essential toolbox for systems of nonlinear equations.

1. Included with

#### Lotka-Volterra I

Learn how to visualize nonlinear pairs with a direction field.

2. Included with

#### Lotka-Volterra II

Apply multivariable calculus ideas to an important pair of nonlinear equations.

3. Included with

#### Linearization

Use linear algebra to figure out the nature of equilibria.

4. Included with

#### The Hartman-Grobman Theorem

Find out why linearization works so well by borrowing ideas from topology.

3. 3

### Partial Differential Equations

The basics of differential equations with many variables.

1. Included with

#### 1D Waves & d'Alembert's Formula

Begin your journey into partial differential equations (pdes) with waves on a length of rope.

2. Included with

#### Sources & Boundary Conditions

What happens when a 1D wave reaches the end of its rope?

3. Included with

#### Challenge: 2D & 3D Waves

Step into higher dimensions by modeling surface ripples and sound waves.

4. Included with

#### Separation of Variables & Waves

Learn how to split a difficult linear pde into a set of easier problems.

4. 4

### Transform Methods

Learn how to turn hard diff eq problems into easy ones.

1. Included with

#### The Fourier Transform

Find out what signals analysis and linear pdes have in common.

2. Included with

#### Practice: Fourier & The Heat Equation

Learn to wield the power of the Fourier transform by practicing on the heat equation.

3. Included with

#### Practice: Fourier & Laplace's Equation

Continue to develop your Fourier transform skills with Laplace's equation.

4. Included with

#### Challenge: Fourier & 3D Waves

Rise to the challenge of modeling 3D compression waves with the Fourier transform.

5. 5

### Power Series

Discover the power of unending sums.

1. Included with

#### Series Solutions I

Learn how infinite sums clear the path to solving difficult differential equations.

2. Included with

#### Series Solutions II

Explore some quirks and special features of infinite series.

3. Included with