Houjun Liu

Differential Equations Index

Differential Equations. math53.stanford.edu.

Logistics

Prof. Rafe Mazzeo

TAs

  • Rodrigo Angelo
  • Zhenyuan Zhang

Assignments

  • Pre-lecture reading + questionnaire
  • PSets: Wed 9A
  • 2 Midterms + 1 Final: wk 4 + wk 7, Thurs Evening; Tuesday 12:15

Review

  1. it suffices to study First Order ODEs because we can convert all higher order functions into a First Order ODEs
  2. homogeneous linear systems \(y’=Ay\) can be solved using eigenvalue, matrix exponentiation, etc. (recall that special cases exists where repeated eigenvalues, etc.)
  3. inhomogeneous systems \(y’ = Ay +f(t)\) can be solved using intergrating factor or variation of parameters method
  4. general analysis of non-linear \(y’=f(y)\): we can talk about stationary solutions (1. linearize each \(y_0\) stationary solutions to figure local behavior 2. away from stationary solutions, use Lyapunov Functions to discuss), or liapenov functions
  5. for variable-coefficient ODEs, we decry sadness and Solving ODEs via power series

Content

What we want to understand:

  • qualitative behaviors and values
  • writing it as an elementary function is lame

Linear ODEs

Linear Second Order ODEs (and how to first-order them)

Non-linear ODEs

Linear Non-Constant Coefficient ODEs

PDEs

Midterm Sheet

Other Stuff