If analytical solvers fail in solving an ODE, does it mean that it is unlikely have an analytical solution?

Question

If analytical solvers (For example, Mathematica and Maple) fail to solve an ODE (Here, I have such situation with the ODE I have given in this question: second order ODE with time-dependent coefficients), does it mean that it is unlikely to have an analytical solution without any approximations to further simplify it?

Answer 1

Maple and Mathematica both contain decision procedures for certain classes of ODEs. The decision procedures all have shape: If ODE $L$ belongs to class X (linear, for example) with coefficients from class Y (frequently, polynomial in the dependent variable), and one is looking for solutions in class Z (polynomial, rational, exp-log, hypergeometric, elementary, etc), then such algorithms will either return an answer or sufficient evidence to prove that there is none.

These algorithms are called by the default de solver routine -- but of course, the default one does not return that evidence of failure in that case. For that, you need to call the exposed in sub-packages decision procedures themselves.

Note that I never said analytical solution anywhere above, as that is ill-defined, as others have pointed out. Are Heun functions allowed in closed-form solutions? People disagree (for good reason) about that. Thus the need for more precision.

Having said that, if neither Maple nor Mathematica find a closed-form solution for you ODE, finding a generic closed-form solution is likely to require very significant work, if it exists at all. Neither are foolproof, but they sure are good.