Comments on: Why Doesn’t This Work? http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/ 12 tables, 24 chairs, and plenty of chalk Tue, 07 May 2013 17:42:45 +0000 hourly 1 http://wordpress.com/ By: TwoPi http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-136 Mon, 24 Dec 2007 14:44:01 +0000 http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-136 I point out that technicality because one obvious way to combine those linear ODEs into a single ODE leads to a nonlinear equation.

I’ve had some fun brainstorming sufficient conditions on the order of each ODE and the associated eigenvalues that would lead naturally to a single linear ODE encompassing both solution spaces. Playing there has me nearly convinced that…oh wait, we’re saving the cool punchlines, aren’t we. :-)

]]> By: Batman http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-132 Sat, 22 Dec 2007 00:37:18 +0000 http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-132 Right, right, yes – linear. I was thinking of the above equations, which are indeed linear, but it’s not true for general homogeneous equations.

]]> By: TwoPi http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-131 Fri, 21 Dec 2007 16:45:07 +0000 http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-131 You’d only get a vector space for a given homogeneous LINEAR differential equation, no? That puts a bit of a crimp in things….

]]> By: Batman http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-129 Fri, 21 Dec 2007 14:39:34 +0000 http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-129 I agree that the problem becomes even more interesting if we permit different orders for f and g. We know that the solutions of a given homogeneous differential equation form a vector space, but when attempting to combine solutions of different equations and find a new equation to satisfy… well, that’s the idea, isn’t it?

]]> By: Ξ http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-127 Fri, 21 Dec 2007 10:17:26 +0000 http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-127 I hadn’t even meant to imply that n is fixed — I think it might be more interesting when the power is allowed to be arbitrary.

]]> By: TwoPi http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-126 Fri, 21 Dec 2007 09:43:50 +0000 http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-126 What a great example!

I wonder if anyone has written about this phenomenon in the vast literature of spurious proofs that 0 = 1.

If you do the general antiderivative, when integration by parts “fails” a less than 100% careful reading of the resulting formula seems to imply that 0 = a nonzero constant. The examples that you commonly see of integration by parts failing lead to *everything* cancelling out, and one often describes that [inaccurately!] as reducing to the equation 0 = 0. I love this example, which doesn’t fit that mold, and points out the sloppiness in drawing the 0=0 conclusion in the other cases.

Ξ’s comment hints at some cool linear algebra content, but even for fixed n, the issue is provocative.

]]> By: Ξ http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-123 Thu, 20 Dec 2007 21:29:23 +0000 http://threesixty360.wordpress.com/2007/12/19/why-doesnt-this-work/#comment-123 It’s funny, this question does seem simple at first but the more I look at it the more maybe-not-earth-shattering-but-certainly-interesting questions pop up.

For example, suppose two functions satisfy:
f^{(n)}(x) = c_{1}f(x) and g^{(n)}(x) = c_{2}g(x).
Under what circumstances will their sum also have a derivative that is a constant multiple of the original sum? What about their product? I want this set of functions to form something nice group- or algebra-wise, but I’m not certain if it does.

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