METODA BISEKCE

F[x_]=Sin[x];

Bisekce[a0_,b0_,m_]:=Module[{a=N[a0],b=N[b0]},
c=(a+b)/2;
k=0;
While[k<m,
If[Sign[F[b]]==Sign[F[c]],b=c,a=c; ];
c=(a+b)/2;
k=k+1; 
];
(*
Print[" c = ", NumberForm[c,16]];
Print[" Δc = ±", (b-a)/2];
Print[" F[c] = ", NumberForm[F[c], 16]];
*)
Return[c];
];

Bisekce[1,4,100]

METODA STŘELBY

F[α_]:=Module[{},
res=NDSolve[{y1'[x]==y2[x],y2'[x]+26 Sin[y1[x]]==0,y1[0]==α,y2[0]==0},{y1[x],y2[x]},{x,0,π}];
vysl=((y2[x]/.res)/.x->π)[[1]];
Return[vysl];
];
Plot[F[α],{α,-π,π}]

y0=Bisekce[0.2,1.0,100]
res=NDSolve[{y1'[x]==y2[x],y2'[x]+26 Sin[y1[x]]==0,y1[0]==y0,y2[0]==0},{y1[x],y2[x]},{x,0,π}];
Plot[Evaluate[y1[x]/.res],{x,0,π},PlotRange->All]

DIAGRAM ŘEŠENÍ

Clear[FF]
FF[α_,λ_]:=Module[{},
res=NDSolve[{y1'[x]==y2[x],y2'[x]+λ Sin[y1[x]]==0,y1[0]==α,y2[0]==0},{y1[x],y2[x]},{x,0,π}];
vysl=((y2[x]/.res)/.x->π)[[1]];
Return[vysl];
];

data={};
For[λ=0,λ<28,λ+=0.1,
F[α_]:=FF[α,λ];
krok=0.1;
For[a=-π,a≤π,a+=krok,
If[F[a]F[a+krok]<0,
vysl=Bisekce[a,a+krok,10];
AppendTo[data,{λ,vysl}]
]
]
]

ListPlot[data,PlotStyle->{PointSize[0.004]},PlotRange->{{0,28},{-π,π}},GridLines->{{0,1,4,9,16,25},{}},AspectRatio->Automatic,PlotLabel->"diagram řešení",AxesLabel->{"λ","α"}]