Question

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 dened on an open set
 R, we say
that the ODE is dened on
. (Why do we take an open set?)

Answer 1

"dened" ?

Answer 2

defined*
both places
sorry about typo

Answer 3

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 \((a,b)\subset\mathbb{R}\).