Wikipedia:Reference desk/Archives/Mathematics/2024 May 23

= May 23 =

Differential Equation
Does the differential equation x2 * d2x/dt2 = k   have a name? (All I've figured out about this so far is that I don't remember enough about differential equations. I'm not getting anything on solving it except errors.)  Thank you. RJFJR (talk) 02:45, 23 May 2024 (UTC)


 * As to a solution, you could guess that one might be some power of $$t$$ and accordingly substitute $$x = A t^{p}$$ (where $$A$$ and $$p$$ are constants), then solve for $$p$$ and then solve for $$A$$. catslash (talk) 09:02, 23 May 2024 (UTC)


 * Then, since nothing in the equation depends on the absolute value of $$t$$, you could apply an arbitrary time-shift to get a slightly more general solution $$x = A (t - t_{0})^{p}$$. catslash (talk) 09:20, 23 May 2024 (UTC)


 * You're right. Thank you. p=2/3, I was not expecting that.  I appreciate it.  RJFJR (talk) 14:14, 23 May 2024 (UTC)
 * This is an autonomous second order equation. If you write $$\ddot x -\frac{k}{x^2}=0$$ and multiply with $$\dot x$$, you find $$\frac{d}{dt}(\frac12 \dot x^2 + \frac k x)=0$$ so $$\frac12 \dot x^2 + \frac k x=C$$ for some constant $$C$$. Solve for $$\dot x$$ and you get a separable first order equation. —Kusma (talk) 14:35, 23 May 2024 (UTC)
 * Thank you. I need to dig out the old textbook and start reading.  RJFJR (talk) 19:25, 23 May 2024 (UTC)
 * catslash's solution is a one-parameter family (indexed by $$t_0$$) of very nice solutions, but in general you should be able to solve the initial value problem for any initial values of $$x$$ and $$\dot x$$, so you'll get a two parameter family. It is easy to show that the solution exists; you can get an implicit formula from Mathematica or other symbolic computation software. —Kusma (talk) 12:17, 24 May 2024 (UTC)
 * Autonomous system (mathematics) gives $$t$$ as a two-parameter function of $$x$$, but this function looks uninvertable except for the choice of the parameter $$C_1$$ which makes it correspond to my guessed solution. catslash (talk) 22:43, 25 May 2024 (UTC)