Wikipedia:Reference desk/Archives/Mathematics/2024 May 23
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May 23
editDifferential Equation
editDoes 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 and accordingly substitute (where and are constants), then solve for and then solve for . catslash (talk) 09:02, 23 May 2024 (UTC)
- Then, since nothing in the equation depends on the absolute value of , you could apply an arbitrary time-shift to get a slightly more general solution . 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 and multiply with , you find so for some constant . Solve for 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 ) of very nice solutions, but in general you should be able to solve the initial value problem for any initial values of and , 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)#Special case: x″ = f(x) gives as a two-parameter function of , but this function looks uninvertable except for the choice of the parameter which makes it correspond to my guessed solution. catslash (talk) 22:43, 25 May 2024 (UTC)
- catslash's solution is a one-parameter family (indexed by ) of very nice solutions, but in general you should be able to solve the initial value problem for any initial values of and , 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)
- Thank you. I need to dig out the old textbook and start reading. RJFJR (talk) 19:25, 23 May 2024 (UTC)