Parametric Dependence 

Plot the dependence of on a parameter a in the differential equation.

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X Plot[pfun[a], {a, -4, 4}, AxesLabel -> {a, \!\( \*SubsuperscriptBox[\(\[Integral]\), \(0\), \(5\)]\(y[s, a] \[DifferentialD]s\)\)}, ImageSize -> Medium]

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X outFun[f_] := ParametricNDSolveValue[eqns, f, {t, 0, 5}, {a}]; dataRow[f_] := Plot[outFun[f][pa], {pa, -3, 3}, AxesLabel -> {a, f}, ImageSize -> Small]; outputs = {{y[1] + y[2], \!\( \*SubsuperscriptBox[\(\[Integral]\), \(0\), \(5\)]\(y[ s] \[DifferentialD]s\)\)}, {y'[3], y[Abs[a]]}};

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X Grid[Map[dataRow, outputs, {2}], Frame -> All, ItemStyle -> {Automatic}, Spacings -> {1.5, 1.5}]

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X eqns = {y''[t] + y[t] == 3 a Sin[y[t]], y[0] == y'[0] == 1}; pfun = ParametricNDSolveValue[eqns, \!\( \*SubsuperscriptBox[\(\[Integral]\), \(0\), \(5\)]\(y[ s] \[DifferentialD]s\)\), {t, 0, 5}, {a}];

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Find the dependence of several other expressions on the parameter a.

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