Date modified: Sun 2025-01-05 . 10:52 AM
Date created: Sun 2025-01-05 . 10:00 AM
# Key ideas Some key ideas I'd like to cover, time permitting. - [ ] Introductory words: What and where do differential equations come from, modeling problems. - [ ] Example. Snowfall - [ ] Using given conditions - [ ] Interlude - Binomial theorem - [ ] Solving when we can apply FTC, for DE of the form $y'(x) = f(x)$. - [ ] Classification - [ ] Ordinary vs partial - [ ] Order of the DE - [ ] Linearity - [ ] Autonomous vs non-autonomous - [ ] What do we mean by a solution - [ ] Interval of definition (a connected interval) - [ ] Family of solutions (described by $k$ many parameters) - [ ] A question: Do solutions even exist? If so, is it unique? - [ ] Example. Picard's warning. - [ ] Interesting aside: Does this contradict Newtonian causality? C.f. Norton's dome. - [ ] Example. Newton law of cooling / heating. Murder mystery. - [ ] Separation of variables, manipulating differential forms. - [ ] Using conditions - [ ] Family of solutions - [ ] Initial value problems. - [ ] Existence and uniqueness theorem (for local solution). - [ ] Magic mapping using iterates to find solutions via convergence - [ ] Picard iterates. - [ ] First order linear ODEs - integrating factor - [ ] Exact differential equations. - [ ] Differentials and potential functions. - [ ] Finding integrating factor for non-exact case. - [ ] Recognition of special differential forms - [ ] Interlude - Euler's formula and trigonometric identities. - [ ] Example. What the curve - A shape that reflects all parallel rays to a single point. - [ ] Orthogonal family of curves. - [ ] Some special techniques - [ ] Recognition of first order DE - [ ] Substitution - [ ] Bernoulli - [ ] Homogeneous ODEs - [ ] Linear substitution - [ ] Stability of equilibria of autonomous systems - [ ] Euler's method - [ ] Programming component. Python / Javascript - [ ] Linear ODEs - [ ]