Linear multiple-variable function I'm reading a differential equation book but i'm stuck on his definition of linear ODE.
Supposing our equation has y as its dependent variable, t as its independent variable and y^(n) reffers to the nth derivative of y with respect to t, my book defines any ordinary differential equation to be of the form :     
$f( y^{(n)},y^{(n-1)},\,\ldots,y',y,t) = 0$   which i clearly agree.     
Then he says that the ordinary differential equation would be called linear iff $f$ were to be linear for $y^{(n)},y^{(n-1)},\,\ldots,y',y$.
But my problem is that it says that if is linear for $y^{(n)},y^{(n-1)},\ldots,y',y$, then:
$$f( y^{(n)},y^{(n-1)},\,\ldots,y',y,t_1) = a(t)y^{(n)} + b(t)y^{(n-1)} + \ldots + c(t)y' + d(t)y + g(t).$$
I'm trying to understand in a simple case.   
Suppose we have a two-variable function $f(x,y)$.
What would the requirement be for $f$ to be linear for $x$?   Would it be that $f(c_1x_1 + c_2x_2,y) = c_1f(x_1,y) + c_2f(x_2,y)$ ?
Or, would it be that $f(c_1x_1 + c_2x2_,y) = f(c_1x_1,y) + f(c_2x_2,y)$?     
Because if I define $f(x,y) = a(y)x + b(y)$  ( like the author from my book defined a multiple-variable linear function )  then it doesn't pass either of the tests ....
Is the definition of a linear function for two-variables completely unrelated to $f(c_1x_1 + c_2x_2,y) = c_1f(x_1,y) + c_2f(x_2,y)$  or $f(c_1x_1 + c_2x2_,y) = f(c_1x_1,y) + f(c_2x_2,y)$?  
Is it completely unrelated to homogeneity and superposition (like in one-variable functions) ?       
What would be the "test" we would make to verify if $f(x,y)$ is linear for $x$ ( other than it fits the form $a(y)x + b(y)$ for instance ) ?        
 A: You are right. What your book means by a linear function is not exactly linear. It's more like an affine function.
Anyway, it's not that important IMHO. The definition :

f( y^(n),y^(n-1),...,y',y,t1) = a(t)y(n) + b(t)y(n-1) + ... + c(t)y' + d(t)y + e(t). 

seems to be more than enough for the scope of your studies (I assume that y is always a function $\mathbb{R} \rightarrow \mathbb{R}$.
EDIT : 
You're absolutely right for your definition of linearity in respect to one variable. See the wikipedia page on multilinear functions (http://en.wikipedia.org/wiki/Multilinear_map), even if your function f is not multilinear, since there's no linearity in respect to t.
Actually I can't recommend a book since I only know french books. But I doubt you'll find a book about this, i feel that this is kind of useless if you want to study linear ODEs where the unknown function is real-valued.  Algebra is useful with linear ODEs, but then you will need to study ODEs with vector-valued unknowns.
At this point, i feel that your book is using the word linear very sloppily. A linear ODE would be defined as : $$g(y^{n}(t), ..., y(t), t) - h(t) = 0$$ where $g$ is linear in respect to every variable except $t$, and $h$ is a function.
