On 26/07/2012 12:51 PM, Invisible wrote:

> Now suppose that by some bizarre mechanism, the spring actually pushes
> the weight in the direction it's already going, rather than back against
> it. Then our differential equation becomes
>
> f''(t) = f(t)
>
> Again, f(t) = 0 is one valid solution. But supposing the weight /does/
> ever move, it seems obvious that it would accelerate forever, without
> limit. And indeed, any elementary calculus textbook will reveal that
> there is precisely /one/ function who's derivative equals the original
> function. This function is exp. So if we write
>
> f(t) = exp t
>
> then it follows that /all/ derivatives of f (including f'') would equal
> f. In other words, this solves our differential equation.

It gets better. I just looked it up, and apparently the derivative of 
sinh is cosh, and the derivating of cosh is sinh again. So we have

   f(t) = sinh(t)
   f''(t) = sinh(t)

Or, equally well,

   f(t) = cosh(t)
   f''(t) = cosh(t)

So the solutions to one equation are sin and cos, and to the other are 
sinh and cosh. That's really pretty...

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