Are all linear second order differential equations in both real and complex spaces solvable by numerical or analytical methods when we are given just the equation provided that a solution exists somewhere in both real and complex spaces.Solution could be of the form of an implicit equation, parametric equation, infinite series, elementary functions, special functions, etc. Will at least one of the methods work in general for linear second order partial differential ntial equations in general?

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    $\begingroup$ If a solution exists then it is by definition solvable, no? $\endgroup$ – user10354138 Feb 23 at 6:39
  • $\begingroup$ DE given in books or asked in exams are doable by hand in terms of simple known functions. Other DEs may or may not be solvable by hand, they may require numerical computations. $\endgroup$ – Z Ahmed Feb 23 at 6:49
  • $\begingroup$ Well just because a solution exists doesn't mean we know what the solution is. $\endgroup$ – Chaitanya Tarkunde Feb 23 at 7:15

The question is ambiguous because the meaning of "solvable" is not precise enough (in fact not defined). Do you mean :

Solvable on the form of an implicit equation ?

Solvable on the form of parametric equation ?

Solvable on the form of an infinite series ?

Solvable in terms of elementary functions ?

Solvable in terms of special functions ?


Moreover an ODE can be not solvable yesterday and be solvable today in terms of special function. For example this is the case of $y''(x)+\frac{1}{x}y'(x)+y(x)=0$ which was not solvable in terms of special functions before the Bessel functions where defined studied and standardized.

Many special functions were standardized in order to make solvable some differential equations https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales .

Thus a today "non-solvable" ODE can become "solvable" tomorow not only because new methods of solving are likely to be invented but simply because new functions are likely to be defined and standardized.

The question is too wide for a definitive answer.

  • $\begingroup$ I mean by all of the above methods $\endgroup$ – Chaitanya Tarkunde Feb 23 at 11:43
  • $\begingroup$ Any type of solution even approximate general method is fine if it exists $\endgroup$ – Chaitanya Tarkunde Feb 23 at 11:43
  • $\begingroup$ The question is too wide for a definitive answer. $\endgroup$ – JJacquelin Feb 23 at 14:22
  • $\begingroup$ I know the question is too "wide" that is why I restricted the order to 2 and asked for a linear equation because those are the most applied ones.(e.g- Schrodinger equation) Also I know about research into pdes as peaple find more ways to solve them.All I ask is if all this is solvable at least in one of methods terms of series, implicit function, special functions, approximations, etc combined. $\endgroup$ – Chaitanya Tarkunde Feb 23 at 15:33
  • $\begingroup$ I have edited my question to be less ambiguous, thanks for the feedback. $\endgroup$ – Chaitanya Tarkunde Feb 23 at 15:47

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