## anonymous one year ago i'm not able to understand this: (this is WRT ODE course.) Linear - When do we say that F(x; y; y1; : : : ; y(n)) = 0 linear? Write F(x; y; y1; : : : ; y(n)) = 0 as L(y)(x) = L(y; y1; : : : ; y(n)) = f(x); and now L is a linear transformation from Cn = the space of functions which are at least n-times dierentiable to F = the space of functions. If these functions are dened on an open set R, we say that the ODE is dened on . (Why do we take an open set?)

1. anonymous

"dened" ?

2. anonymous

defined* both places sorry about typo

3. anonymous

I believe the "open set" requirement is there for the same reason a function $$f:\mathbb{R}\to\mathbb{R}$$ can only be differentiable over an open interval $$(a,b)\subset\mathbb{R}$$.