Just a simple question:
Is cosxdy/dx = x non-linear because of cosx?
An differential operator $\displaystyle $$ L$ is linear iff for any two solutions $\displaystyle y_1(x)$ and $\displaystyle y_2(x)$ of $\displaystyle Ly=0$ and any constants $\displaystyle \alpha, \beta \in \mathbb{C}$ then $\displaystyle y(x)=\alpha y_1(x)+\beta y_2(x)$ is also a solution. An ODE is linear if it is of the form $\displaystyle Ly=f(x),\ L$ a linear differential operator
Now you can check for yourself.
(the answer is: it's linear)
CB
According to the wiki, a linear differential equation is of the form
$\displaystyle Ly(x)=f(x),$ where $\displaystyle L$ is a linear differential operator.
That is, you apply CB's definition of linearity to the operator, not necessarily to the entire equation. If the equation is homogeneous, they'll turn out to be the same thing. But even a non-homogeneous linear DE does not obey the superposition principle except in its homogeneous solution.
This is my understanding.