Aufgabe 1

F=m0 c^2 (1/Sqrt[1-v^2/c^2]-1)

Ekin=Series[F,{v,0,4}]

Limit[Normal[Ekin],c-> Infinity]

Fehler:

Ekin-%

Aufgabe 2

Erste Alternative:

(a)

F0=(((((1+x)/(1-x))))^α)

F1=D[F0,x]

Together[(F1/F0)]

Dami erhält man

D[F[x],{x,1}]+((2α)/((-1+x)(1+x)))F[x]==0

DE=Numerator[Together[D[F[x],{x,1}]+((2α)/((-1+x)(1+x)))F[x]]]

Collect[DE,D[F[x],{x,_}],Factor]==0

(b)

F0=Exp[-(1/(x^2))]

F1=D[F0,x]

DE=D[F[x],x]-x^3/2 F[x]

Numerator[Together[DE]]==0

(c)

F0=Cos[x]Sin[x]

F1=D[f0,x]

Together[F1/F0]

F2=D[F1,x]

Clear[A]
ansatz=A[0]*F0+A[1]*F1+A[2]F2

sol =Solve[{ansatz[[1]]+ansatz[[2]]==0,ansatz[[3]]==0},{A[0],A[1],A[2]}]

DE=A[0]*F[x]+A[1]*D[F[x],x]+A[2] D[F[x],{x,2}]/.sol[[1]]

DE/.A[2]->1

(d)

F0=((ArcSin[x])^2)

F1=D[F0,x]

F2=D[F1,x]

F3=D[F2,x]

Together[F2/F1]

Clear[A];
ansatz=A[0]*F0+A[1]*F1+A[2]*F2+F3

Folgendes Gleichungssystem bekommen wir

G1=A[0]((ArcSin[x])^2);
G2=A[2] (2/(1-(x^2)))+((6 x)/(((1-(x^2)))^2));
G3=A[1]((2 ArcSin[x])/(Sqrt[1-(x^2)]))+A[2]((2 x ArcSin[x])/(((1-(x^2)))^(3/2)))+((6 (x^2) ArcSin[x])/(((1-(x^2)))^(5/2)))+((2 ArcSin[x])/(((1-(x^2)))^(3/2)));
sol = Solve[{G1==0,G2==0,G3==0},{A[0],A[1],A[2]}];

sol=Map[Together,sol]

DE=(∑_(k=0)^3)A[k]*D[F[x],{x,k}]/.sol[[1]]/.A[3]->1

Collect[Numerator[Together[DE]],D[F[x],{x,_}]]==0

DE=F'[x]+3 x F''[x]+(-1+(x^2)) Derivative[3][F][x]

Collect[DE,D[F[x],{x,_}], Factor]==0

(e)

F0=((p[x])/(q[x]));
F1=Together[D[F0,x]]

DE=D[F[x],x]-((q[x] p'[x]-p[x] q'[x])/(q[x]p[x]))F[x]

DE1=Numerator[Together[DE]]

Collect[DE1,D[F[x],{x,_}], Factor]==0

Zweite Alternative:

MAXDEORDER=5

Clear[HolonomicDE];
HolonomicDE[f_,F_[x_],print_:False]:=Module[{DF,eqcl,rat,A,approach,sol,eq,i,j,l,m},
DF=Table[D[f,{x,i}],{i,0,MAXDEORDER-1}];
DF=Map[If[Head[#]===Plus,Apply[List,#],{#}]&,DF];(* DF consists of lists of derivatives (sums are lists of summands) *)
eqcl={A[1]DF[[1]]};
For[i=2,i≤MAXDEORDER,i++,
For[l=1,l≤Length[DF[[i]]],l++, (* for every summand of the (i-1)-th derivative do the following *)
m=Length[eqcl];
For[j=1,j≤m,j++,
rat=Together[DF[[i,l]]/eqcl[[j,1]]]; (* building quotient of l-th summand and first element of j-th eq. class*)
If [PolynomialQ[Denominator[rat],x] && PolynomialQ[Numerator[rat],x], (* test, if rat is rational or not  *)
AppendTo[eqcl[[j]],A[i]DF[[i,l]]]; (* add to equivalence class *)
Break[],
If[j==Length[eqcl],AppendTo[eqcl,{A[i]DF[[i,l]]}];  (* build new equivalence class *)
];
];
];
];
var=Table[A[k],{k,i}];
approach=Map[Apply[Plus,#]&,eqcl]; (* sum over all elements in equivalence classes *)
approach=Map[#==0&,approach];
If[print,Print[{eqcl,approach}]];
Off[Solve::"svars"];
sol=Solve[approach,var];
If[sol=!={{}} && Union[var/.sol[[1]]]=!={0},
eq=Sum[A[k+1]D[F[x],{x,k}],{k,0,i-1}]/.sol[[1]];
eq=Numerator[Together[eq]]/.A[_]->1;
Return[Collect[eq,D[F[x],{x,_}],Factor]==0]];
];
]

(a)

HolonomicDE[((1+x)/(1-x))^α,F[x],True]

(b)

HolonomicDE[Exp[-1/x^2],F[x],True]

(c)

HolonomicDE[Cos[x]Sin[x],F[x],True]

(d)

HolonomicDE[ArcSin[x]^2,F[x],True]

(e)

HolonomicDE[(a x^2+b x+c)/(x+1),F[x],True]