(adsbygoogle = window.adsbygoogle || []).push({}); The problem statement, all variables and given/known data

Now that I'm working it out as I'm typing it, I think I may have solved this problem; could still be a mistake though...

This was an extra credit problem on the final for my into to diff eq class. I never saw anything like it before and I didn't finish it because I got stuck on one part. It's copied pretty much exactly as it was on the paper.

Define [x] by

[2] = 5

[3] = 8

[5] = 14

[-2] = -7

...

and

[x]^{1}= [x]

[x]^{2}= [[x]]

[x]^{3}= [[[x]]]

....

Solve dy/dx = [10]^{x}

The attempt at a solution

I see that [x] is basically the function f(x) = 3x - 1. Then I can find a formula for [x]^{1}, [x]^{2}, ... by plugging f(x) into itself and finding a pattern.

[x]^{1}= 3x - 1

[x]^{2}= 9x - 4

[x]^{3}= 27x - 13

[x]^{4}= 81x - 40

...

Then I got [x]^{n}= 3^{n}x - (3^{n}- 1)/2

Finding a closed form for the last term of the nth term of [x]^{n}had me stuck and that was where I originally stopped on the problem. I used Wolframalpha to find a closed form, but is there a clever way of finding it just by looking at the numbers, without knowing the formula beforehand? I can't see it...

Now if g(x) = [10]^{x}, then g(x) = 3^{x}(10) - (3^{x}- 1)/2 and I can integrate that to solve dy/dx = [10]^{x}.

Correct?

Something that kind of bothered me is that [10]^{x}looks like it can only be defined for natural numbers, so does dy/dx = [10]^{x}even make sense in the first place?

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# Homework Help: Kinda weird Diff Eq problem

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