Math 53 — Differential Equations Review
A concept-per-page review of Stanford Math 53 (Spring 2026) — first-order ODEs through Fourier transforms. Dark theme, short pages, lots of worked problems.
Part I — Explicit & Qualitative Methods
First-order ODEs, stability, second-order linear ODEs, and linear systems via eigenvalues.
- What is an ODE? · Ch 1
- Order, Linearity, and Solutions · Ch 1
- Initial Value Problems · Ch 1
- Slope (Direction) Fields · Ch 1
- Autonomous Equations · Ch 2
- Separable Equations · Ch 2
- Existence and Uniqueness · Ch 2
- Qualitative Analysis & Long-Term Behavior · Ch 3
- Phase Line & Stability · Ch 4
- Linear Stability Test · Ch 4
- Complex Numbers Refresher · Ch 5
- Second-Order Homogeneous Linear ODEs · Ch 6
- The Characteristic Equation (2nd order) · Ch 6
- Linear Systems via Eigenvectors · Ch 7
- The Matrix Exponential · Ch 8
- 2D Systems: Node / Spiral / Saddle / Center · Ch 9
Part II — Higher-Order & Nonlinear
Inhomogeneous equations, nonlinear systems, chaos, linearization, and conserved quantities.
Part III — Series & Numerical Methods
Power-series solutions, Euler's method, Runge–Kutta, stiffness.
Part IV — PDEs & Fourier Series
Heat equation, separation of variables, Fourier series on intervals.
Part V — Fourier Transform
Exponential Fourier series, Fourier transform, Gaussians, convolution, wave equation on the line.
Built from the Stanford Math 53 textbook and Spring 2026 course materials. See About for sources and scope.