I'm learning about differential equations. I'm starting off with the MIT OCW course, but I'm going to incorporate other sources as well.

Here are some high-level notes that I'll jot down as I go through my self-made curriculum.

Here's some of the high-level stuff that I learned from the MIT course:

• Lecture 1: The Geometrical View
  • Solutions are functions, not numbers. Reminds me of ML.
  • Solutions can be found by integrating over the direction field. This is Euler's method.
• Lecture 2: Numerical Methods
  • Euler's method has O(h) error.
  • RK2 has O(h^2) error.
  • RKn has O(h^n) error.
• Lecture 3: First-order Linear ODEs
  • Always admit a solution (involves an integral, though).
  • "Integrating factors" are the general way to solve these problems.

Here's some of the high-level stuff that I learned via my own derivations or thinking:

• Numerical methods
  • Error analysis can be performed via Taylor series.
    • The error on one step of Euler's method is O(h^2). That's why k/h steps is O(h).
    • The error on one step of RK2 is O(h^3).
  • New methods can be derived using Taylor series and solving a system to knock out the low-order terms:
    • I derived a third-order method, which I believe is a variant of RK3.
    • I derived a fourth-order method that requires 6 function evals. See derivations/fourth_order_*.jpg.