My mother is teaching a high school course on multivariable calculus, and they were studying linear differential equations of the form $$y' + P(x) y = Q(x),$$ and the question of why this equation is called "linear" came up.
In terms that these students are familiar with, since they haven't been exposed to linear algebra yet, my thought was to say that the equation for a line, $y = mx+b$, is "linear" in $x$ (ignoring the technicality that it's actually an affine equation, not a linear one), because it's in the form "coefficient times $x$", and then we allow another term which is just a lonely coefficient. And then we extend this notion to saying that the above differential equation is "linear" in $y$ and $y'$, but this time the coefficients are allowed to be functions of $x$.
That's probably a good enough hand-wavy explanation to help students remember the definition, at the very least. I couldn't really think of a good reason why it "should", a priori, be kosher to allow coefficients to be functions of $x$ here. At that point it seems to me like you just have to get into the linear algebraic definition of linearity, which, being completely foreign to the students... it just seems to be a bit too deep of a rabbit hole for this purpose.
So my question is: does anyone have a better way of approaching this? And if you think my hand-wavy explanation above is largely acceptable, is there a way you can explain why multiplying by non-constant functions of $x$ "should" still be considered linear in $y$?