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Clear["Global`*"]
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x[p_,q_]:=4.`p+Log188.58217139771662`0.7580658066983395`+1.`Tanh2.1332922094779003`+0.0005716925300367255`Log;
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R[p_,q_]:=11+11.810832292494606`;
2
Sech1.1354604292040422`-3.81874`+p+Log9.322865507870436`*^17
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tm[q_]:=Log;
1.433398163769229`*^-11
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y0[q_,r_]:=Log5.5475950225106745`r39.7685`-10.4018`1.`-1.`Tanh1.1354604292040422`-3.81874`+Log9.322865507870436`*^17;ft[Y_,p_,q_]:=Module[{y,b,xx,A,dydt,dbdt},{y,b}=List[Y[[1]],Y[[2]]];xx=x[p,q];A=Sqrt[Exp[y]+Exp[xx]];dydt=(1.+3.*A)/(A-Exp[y-xx]/4.)/R[p,q];dbdt=-A/(A-Exp[y-xx]/4.)/R[p,q];{dydt,dbdt}];BE[q_,r_]:=Module[{yini,bini,Yini,tauM,NT,ty,sol,ys,bs,xs,H2,bh7,H2p,bsp,bspp},yini=y0[q,r];bini=0;Yini={yini,bini};tauM=tm[q];NT=10000;ty=Table[i,{i,0,tauM,tauM/NT}];sol=NDSolve[{Y'[tau]ft[Y[tau],tau,q],Y[0]Yini},Y,{tau,0,tauM}][[1]];ys=sol[[1,All,1,2]];bs=sol[[1,All,2,2]];xs=x[ty,q];H2=Table[Log[Exp[ys[[i]]]+Exp[xs[[i]]]],{i,1,NT}];bh7=Table[Exp[ys[[i]]],{i,1,NT}];H2p=Table[(H2[[i+1]]-H2[[i]])/(ty[[i+1]]-ty[[i]]),{i,1,NT-1}];bsp=Table[(bs[[i+1]]-bs[[i]])/(ty[[i+1]]-ty[[i]]),{i,1,NT-1}];bspp=Table[(bsp[[i+1]]-bsp[[i]])/(ty[[i+1]]-ty[[i]]),{i,1,NT-2}];{ty,bh7,H2,H2p,bs,bsp,bspp,NT}];
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BE[1,1]
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$Aborted[]