# Tricky equation

1. Dec 31, 2007

### lewis198

I was wondering what series approximation I can use to approximate y:

y=(1-(dx/dy)^2)^1/2

when dx/dy is not trigonometric, and contains the derivative of a floor function

2. Dec 31, 2007

### dodo

Hi,
the floor function is notoriously discontinuous. What do you mean by its derivative?

3. Dec 31, 2007

y has properties of sin fuction i guess we solve x in terms of y

x=1/2*(abs(y)(1-y^2)^0.5 +inverse sin(abs(y)))

you can also get it solved for y

Last edited: Dec 31, 2007
4. Jan 1, 2008

### lewis198

okay well an approximation

5. Jan 1, 2008

what do you meant by that

6. Jan 6, 2008

### epenguin

You can transform that into dx/dy = sqrt(1 - y^2) and integrate wrt y to obtain

x = -.5 {y*srt(y^2 - 1) - ln ABS(y + sqrt(y^2 -1)} + a constant

give or take a + or -

which does not mean the solution of sadhu is not right too.

Did this d.e. emerge from any 'real' problem?

7. Jan 6, 2008

### olgranpappy

how about $\sum_n \delta(x-n)$.

8. Jan 6, 2008

### Gib Z

I don't exactly know what you mean by that :( But I doubt that's a derivative. Differentiability implies continuity. The floor function, being discontinuous at an infinite number of points, is not differentiable.

9. Jan 7, 2008

### HallsofIvy

Staff Emeritus
That uses the delta "function". It's not a function in the true sense, but a distribution or "generalized function". And, of course, the differentiation is in the sense of distributions. Distributions do not have to be continuous in order to be differentiable. (In fact, I am not sure that "continuous" is defined for distributions!)

10. Jan 7, 2008

### olgranpappy

$\delta$ is Dirac's delta function and the $n$ are positive integers. You can, of course, check for yourself that
$\int_0^x dy \sum_n \delta(y-n)=Floor[y]$

Lah dee dah, lah dee dah, I'm not worried about that...

But, if it's not differentiable then the derivative doesn't exist. So how did I just write the derivative down if it doesn't exist?... I'd rather not let the fact that something doesn't exist stop me from using it to solve a problem.

Mathematicians are so cute...

11. Jan 8, 2008

### Gib Z

I was never stopping you from solving the problem. Just don't call it a "derivative". Call it something else. You didn't write down the derivative, you wrote down something that satisfies a nice equation in this same manner a derivative does for continuous functions. That doesn't mean it is a derivative.

Physicists are just sloppy. There is nothing in mathematics that stops the physics being done, but physicists are too lazy to justify their mathematics. Every working physicist I have asked admits that their sloppy mathematics leads them to errors, why not correct the problem?

12. Jan 8, 2008

### olgranpappy

nah.

13. Jan 8, 2008

### morphism

It's simple: you didn't.

14. Jan 8, 2008

### olgranpappy

oh, girls, stop being so silly.