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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)[reply]

As to a solution, you could guess that one might be some power of

t

and accordingly substitute

x=At^{p}

(where

A

and

p

are constants), then solve for

p

and then solve for

A

. catslash (talk) 09:02, 23 May 2024 (UTC)[reply]

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)[reply]

You're right. Thank you. p=2/3, I was not expecting that. I appreciate it. RJFJR (talk) 14:14, 23 May 2024 (UTC)[reply]

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}}({\frac {1}{2}}{\dot {x}}^{2}+{\frac {k}{x}})=0

so

{\frac {1}{2}}{\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)[reply]

Thank you. I need to dig out the old textbook and start reading. RJFJR (talk) 19:25, 23 May 2024 (UTC)[reply]

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)[reply]

Autonomous system (mathematics)#Special case: x″ = f(x) 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)[reply]