- #1

Eclair_de_XII

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## Homework Statement

This is the definition I got from my book:

"A crucial classification of differential equations is whether they are linear or nonlinear. The ordinary differential equation

##F(t,y,y',...,y^{(n)})=0##

is said to be linear if F is a linear function of the variables ##y, y',...,y^{n}##; thus the general linear ordinary differential equation of order ##n## is

##a_0(t)y^{(n)}+a_1(t)y^{(n-1)}+...+a_n(t)y=g(t)##

## Homework Equations

I saw something similar in my Linear Algebra class used to prove linear independence that dictates the conditions needed, that none of the vectors [derivatives] can be written as linear combinations of each other. Basically a set of vectors ##v_i## is a linearly independent set if the below equation is satisfied for any non-zero constant ##a_i##.

##a_1v_1+...+a_nv_n=0##

##a_0(t)y^{(n)}+a_1(t)y^{(n-1)}+...+a_n(t)y=g(t)=0##

I think it'll probably come up in my Linear Algebra/Differential Equations class I will eventually be taking, but this was the closest relevant equation I could relate the concept of linear differential equations to.

## The Attempt at a Solution

Anyway, my conclusion is that the ##a_i(t)## are all constant

*relative*to the independent variable ##y## (that is, they do not change as the value of ##y## or any of its derivatives change for any ##t##. The ##a_i(t)## can still be non-linear in respect to ##t##, but not to ##y##. Can anyone provide for me a better explanation of why a coefficient that is a function of ##t## can be nonlinear?