Enough of politics and cats. Time to blog about math and physics again.

Back in my high school days, when I was becoming familiar with calculus and differential equations (yes, I was a math geek) something troubled me. Why were certain expressions called “linear” when they obviously weren’t?

I mean, an expression like $$Ax+B$$ is obviously linear. But who in his right mind would call something like $$x^3y + 3e^xy+5$$ “linear”? Yet when it comes to differential equations, they’d tell you that $$x^3y+3e^xy+5-y^{\prime\prime}=0$$ is “obviously” a second-order, linear ordinary differential equation (ODE). What gives? And why is, say, $$xy^3+3e^xy-y^{\prime\prime}=0$$ not considered linear?

The answer is quite simple, actually, but for some reason when I was 14 or so, it took a very long time for me to understand.

Here is the recipe. Take an equation like $$x^3y+3e^xy+5-y^{\prime\prime}=0$$. Throw away the inhomogeneous bit, leaving the $$x^3y+3e^xy-y^{\prime\prime}=0$$ part. Apart from the fact that it is solved (obviously) by $$y=0$$, there is another thing that you can discern immediately. If $$y_1$$ and $$y_2$$ are both solutions, then so is their linear combination $$\alpha y_1+\beta y_2$$ (with $$\alpha$$ and $$\beta$$ constants), which you can see by simple substitution, as it yields $$\alpha(x^3y_1+3e^xy_1-y_1^{\prime\prime}) + \beta(x^3y_2+3e^xy_2-y_2^{\prime\prime})$$ for the left-hand side, with both terms obviously zero if $$y_1$$ and $$y_2$$ are indeed solutions.

So never mind that it contains higher derivatives. Never mind that it contains powers, even transcendental functions of the independent variable $$x$$. What matters is that the expression is linear in the dependent variable. As such, the linear combination of any two solutions of the homogeneous equation is also a solution.

Better yet, when it comes to the solutions of inhomogeneous equations, adding a solution of the homogeneous equation to any one of them yields another solution of the inhomogeneous equation.

Notably in physics, the Schrödinger equation of quantum mechanics is an example of a homogeneous and linear differential equation. This becomes a fundamental aspect of quantum physics: given two solutions (representing two distinct physical states) their linear combination is also a solution, representing another possible physical state.