How to explain to a high school student why a linear differential equation is linear?

Question

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 $y$?

Answer 1

The differential equation $y'+Py=Q$ is linear because the underlying homogeneous problem $y'+Py=0$ satisfies the linearity property that when $y_1, y_2$ are solutions the arbitrary linear combination $c_1y_1+c_2y_2$ is once more a solution. It is linearity of the solution set rather than the explicit form of the differential equation itself which is of interest. Of course, there is also the superposition principle, if we replace $Q$ with $c_1Q_1+c_2Q_2$ then the solutions of $y'+Py=Q_1$ and $y'+Py=Q_2$ superpose to give solutions of $y'+Py=c_1Q_1+c_2Q_2$ hence the net-cause is a sum of the individual effects. All of these features are characteristic of linear systems.

Answer 2

I think the best way might be to think about what would happen if we had two solutions to this $y_1$ and $y_2$. Then, we could consider that, for $a+b=1$, the function $ay_1+by_2$ is also a solution. It's rather obvious that this is a line if they happen to know linear algebra, but otherwise, here's how I'd explain it:

Draw the plane with axes $a$ and $b$ and the line $a+b=1$ - which the students should be able to recognize as a line given what is taught in high school. Notice that this passes through the points $(1,0)$ and $(0,1$). Consider that, if $(a,b)=(1,0)$, then $ay_1+by_2=y_1$ - so the point $(1,0)$ on the coordinate plane represents the function $y_1$. Similarly, the point $(0,1)$ represents the function $y_2$. Then, as an example, the middle point $\left(\frac{1}2,\frac{1}2\right)$ can be viewed as the average $\frac{y_1+y_2}2$, which should be another familiar form. You could also look at other points as weighted averages. Doing this should give them a sense that this line in the $ab$ coordinate plane represents that any function in some sense "between" or arising from an average of two others must also be a solution - and the students, as a bonus, have the visual of a line on a plane, which, at the very least, acts as a mnemonic if not to give insight.

Answer 3

Just tell them that the word "linear" (when used outside of elementary algebra) is defined to mean "homogeneous of degree 1" and "additive". Give an example of each of those 2 properties, explain that it's really the homogeneous equation that's linear, and move on. It's just a definition, and they'll get a better idea of it when they get to linear algebra so there's no reason to waste time on it.

/rant Although, I should say that taking multivariable or ODEs without first having had linear algebra is not a great idea. I wish high schools would quit doing this. The ideas of multivariable calculus and ODEs can sink in a LOT deeper, if one isn't starting from scratch. And because of the way AP credit works, the students may be moving on to even higher level math courses (PDEs, etc) immediately after they get to college without ever really understanding what they were doing with all those Jacobians, tangent spaces, and Wronskians. /endrant

Answer 4

As Nameless pointed out in his comment, talking about what does Gross' "$L$-machine" do to linear combinations of functions can be useful and easy to understand, given that students already know derivatives properties:

$$ L(y) = y'+P(x)y,$$

$$ L(ay_1+by_2) = aL(y_1)+bL(y_2).$$

The derivative machine ($y'$) is already known to be "linear". We are expanding this machine carefully (+$P(x)y$) in a way in which we don't lose that "linearity".