Differential equation I can't solve

[Irid]

Hi,
I was solving my own-invented physical problem, and obtained a differential equation
$$\frac{dx}{dt} = k \sqrt{t-x^2}$$
with k positive constant. I wish to solve it. I'm unable perform separation of variables here, and power-series method works poorly, too. I conclude that this equation does not have any closed-form or familiar series solution, so the best thing that I could do is use numerical methods [of an Excel type, but Gnumeric is my choice :)]. It turned out that with k=1, when t=1, x=0,6, exacty. This gives a clue that there might exist some sort of a solution (or it might not be any kind of a clue, I'm just being silly). How do I find it? How do I solve this equation?

 

[Avodyne]

There's no exact solution. I'm assuming you're using a boundary condition of x=0 at t=0. Then, Mathematica's numerical routine NDSolve gives x=0.60635 at t=1.

You can solve it approximately by Taylor series for small t and large t. At small t,
$$x = {{\textstyle{2\over3}}t^{3/2}\left(1-{\textstyle{2\over21}}t^2 +{\textstyle{10\over2079}}t^4 + \ldots\right)$$
At large t,
$$x = t^{1/2}\left(1-{\textstyle{1\over8}}t^{-2} -{\textstyle{13\over128}}t^{-4} + \ldots\right)$$
These cross at t=2.354, where they differ from the numerical solution by a worst-case 1%.

This is for k=1. But if $$x=f(t)$$ solves the equation for k=1, then $$x=k^{-1/2}f(kt)$$ solves it for general k.