Timeline for Solving $0=-\lambda \phi (t)^3+\mu ^2 \phi (t)+\phi ''(t)$
Current License: CC BY-SA 4.0
11 events
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| Apr 21, 2022 at 14:14 | comment | added | Michael E2 | Yes, yours is the one with the weaker hypotheses that I mentioned. It's also clear that since the indicated form has one parameter, it should be possible (mathematically, if not computationally) to arrive at the form using all but one of the initial conditions. Furthermore, since the parameter corresponds to the autonomous symmetry, it's clear mathematically that your approach is the one to try. (+1) OTOH, I thought from the start that the problem was to be solved without using knowledge of the solution. | |
| Apr 21, 2022 at 8:05 | history | edited | Ulrich Neumann | CC BY-SA 4.0 |
added 845 characters in body
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| Apr 21, 2022 at 7:14 | comment | added | Ulrich Neumann | @MichaelE2 In contrast to Bob's answer (pair of consistent initial conditions) I only need one initial condition ! | |
| Apr 20, 2022 at 17:12 | comment | added | Diffycue | @MichaelE2 Thank you for the input, I have understood that my initial question was unclear and have made the appropriate modification. | |
| Apr 20, 2022 at 16:59 | comment | added | Michael E2 | @Diffycue Yes, it is. Both are helpful. OTOH it is not hard to see that one is based on weaker hypotheses. It's also possible to solve the problem without any hypotheses. -- Ulrich is right to point out autonomous, which implies the order may be reduced as he showed. -- What's hard to see is why ask for a solution you already know? But motivations don't matter a lot to me. | |
| Apr 20, 2022 at 16:54 | comment | added | Diffycue | @MichaelE2 One answer is helpful and the other is not. Is it so hard to see? | |
| Apr 20, 2022 at 16:43 | comment | added | Michael E2 |
@Diffycue So when Ulrich uses the solution to derive the solution it's ok, but not Bob? (Ulrich uses \[Phi][0]==0, without which we get the general solution in terms of theta functions.)
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| Apr 20, 2022 at 16:34 | comment | added | Ulrich Neumann | Perhaps "Separation of variables" but I'm unsure. See Autonomous in MathWorld/Weisstein | |
| Apr 20, 2022 at 16:31 | comment | added | Diffycue | Clever! Is there a name for this technique-- integrating factor or some such? | |
| Apr 20, 2022 at 16:30 | vote | accept | Diffycue | ||
| Apr 20, 2022 at 16:28 | history | answered | Ulrich Neumann | CC BY-SA 4.0 |