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I've always wondered why does the differential equation notation for linear equations differ from the standard terminology of vector spaces.

We all know that the equation $y'' + p(x) y' + q(x)y = g(x)$ for some function $g$ is called linear and that the associated equation $y'' + p(x)y' + q(x) y = 0$ is called homogeneous. But why is that? WHY should mathematicians explicitly cause confusion with the rest of the theory of vector spaces?

What I mean by that is : Why not call the equation $y'' + p(x)y' + q(x)y = g(x)$ an affine equation and call $y' + p(x) y' + q(x) y = 0$ a linear equation? Because linear equations (in the sense of differential equations) are not linear in the sense of vector spaces unless they're homogeneous ; and linear equations (in the sense of differential equations) remind me more of a linear system of the form $Ax = b$ (which is called an affine equation in vector space theory) than of a linear equation at all.

Just so that I made myself clear ; I perfectly know the difference between linear equations in linear algebra and linear equations in differential equations theory ; I'm asking for some reason of "why the name".

Thanks in advance,

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    A similar confusion exists for functions too: a function of the form $f(x) = ax + b$ is often referred to as a *linear function*, even though it is an *affine function*. The reason, as we all know, is that its graph is a straight line. So if I should venture a guess as to what the answer to your question is, it would be that this terminology somehow bled over to the theory of differential equations. Furthermore, this may have happened a fairly long time ago - long enough for the terms to become sufficiently ingrained, so that they're impossible to change at this point.2012-02-12
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    For functions it makes sense to say that $f(x) = ax + b$ is linear *mostly* because we don't say linear meaning linear equation, but we say linear meaning straight line. I don't think we confuse anyone with this. But in differential equations the confusion is often there ; it can even be exam questions, and I must say having taken a course in differential equations and going to the exam without reading my definitions, the teacher asked if some linear equation was linear and I said "no, because it's an affine one" and I got it wrong. This was a long time ago, I'm not raging about this exam...2012-02-12
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    Now I'm just wondering why confusion is being maximized.2012-02-12

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