Question

Weird differential equation

Answer 1

\[\LARGE \frac{d^{ \small ( (\lfloor x \rfloor \mod 2 )+1)} y }{dx^{ \small( (\lfloor x \rfloor \mod 2 )+1)} }=x\]

Answer 2

Isn't \(\lfloor x\rfloor \text{mod }2=1\)?

Answer 3

So the ODE is really
\[y''=x~~\implies~~y=\frac{1}{6}x^3+C_1x+C_2\]

Answer 4

Oh hold on, if \(x=0\) then \(\lfloor x\rfloor\text{mod }2=0\)... Hmm...

Answer 5

Weird that W|A is suggesting it's equal to 1: http://www.wolframalpha.com/input/?i=Floor%5Bx%5D+mod+2

Answer 6

Well in my mind it seems like the derivative is stepping back and forth between the first and second derivative. So it starts out from 0 to 1 as y'=x then 1 to 2 is y''=x then goes back to y'=x from 2 to 3, etc... and just keeps going like that. Since I came up with this nonsense, I just sorta said that we could make a piecewise function where we sort of just cut them together to make it continuous haha.

Answer 7

I don't know about whether or not one can define a DE with this kind of alternating pattern, much less whether we can have the order of the derivative depend on the independent variable. \(y^{\lfloor x\rfloor}=f(x)\) is itself an interesting idea. I'm not sure how one would physically interpret that sort of phenomenon (that is, if it's at all a concrete one).

Answer 8

An interesting question nonetheless

Answer 9

I don't see why we can't do it, whether or not it's real has never concerned a mathematician before. All we're doing is saying the differential equation alternates between being a description of the slope to a description of the concavity. Maybe that could model something like how light passes through a grating, every other part is distinctly covered or not, or maybe water running off of a pyramid from square stone to the next. I don't know just making up stuff that seems plausible and fun.