In this course, you will :
- Discover how to draw nonlinear pairs with a direction field.
- Apply multivariable calculus concepts to a key pair of nonlinear equations.
- Learn why linearization works so well by borrowing topological ideas.
- Learn how to break down a difficult linear pde into a series of simpler problems.
- Learn about the similarities and differences between signal analysis and linear pdes.
- Practice using the Fourier transform on the heat equation to learn how to use it effectively.
- Discover how infinite sums make it easier to solve difficult differential equations.
- Investigate some of the peculiarities and unique characteristics of infinite series.
- What's a good way to tell power series apart?
1. Nonlinear Equations
- The Hartman-Grobman Theorem
2. Partial Differential Equations
- 1D Waves & d'Alembert's Formula
- Sources & Boundary Conditions
- Challenge: 2D & 3D Waves
- Separation of Variables & Waves
3. Transform Methods
- The Fourier Transform
- Practice: Fourier & The Heat Equation
- Practice: Fourier & Laplace's Equation
- Challenge: Fourier & 3D Waves
4. Power Series
- Series Solutions
- The Airy Equation
- Interlude: Return of the Wronskian