(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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Kinda weird Diff Eq problem

**Physics Forums | Science Articles, Homework Help, Discussion**