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Get the admissible set of a parameter in differential equation

Posted 3 months ago
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Hello all,

I am trying to find an "easy way" of getting the possible values for a "a" parameter in the following differential equation:

solutions = ParametricNDSolveValue[{y''[t] - ArcTan[y[t]] + y'[t]^2 == a,  y[0] == y[1] == 0}, y, {t, 0, 1}, {a}]

It's easy to plot them as following, for instance:

Plot[Evaluate[Table[solutions[a][t], {a, -4, 5, .5}]], {t, 0, 1},  PlotRange -> {{0, 1.1}, {-0.6, 0.6}}]}

and also to manipulate and visualize the "a" parameter doing:

Manipulate[Plot[solutions[a][t], {t, 0, 1}, PlotRange -> {{0, 1}, {-1, 1}}],  {a, -4.1, 5.2414, 0.000001}]

But what I really need is to find a way to get a defined set of the admissible "a" parameter in the equation, instead of a graphic visualization.

Can please somebody help showing a suggestion or a way to do it?

Thank you very much! Rui.

Many ODEs do not have symbolic solutions in terms of commonly defined functions.

The ODE in the question probably falls in that category.

However, it should be noted that DSolve sometimes is unable to solve ODEs that do have symbolic solutions in terms of commonly defined functions.

 DSolve[{y''[t] - ArcTan[y[t]] + y'[t]^2 == a}, y[t], t]
 (*Solution with parametric form,with unsolvable integrals *)
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