Lösungen

1

a

Remove["Global`*"];

f[x_,y_]:= 3*x^2 + 9*x^2*y + 2*x*y^3;
g[x_,y_]:= 1 + 3*x^3 - 2*y + 3*x^2*y^2;

y'[x]==-f[x,y[x]]/g[x,y[x]]  (* D'Gl. *)

DSolve[y'[x]==-f[x,y[x]]/g[x,y[x]],y[x],x ]

solv1=DSolve[y'[x]==-f[x,y[x]]/g[x,y[x]],y[x],x ]//N

z[x_]:=y[x]/.solv1[[1]]

z[x]/.C[1]->c

Plot[Evaluate[Table[z[x]/.C[1]->c,{c,-5,5}]],{x,1,10}];

(*Gesucht ist nur die reelle Lösung! *)

b

Remove["Global`*"];

f[x_,y_]:= 3*x^2 + 9*x^2*y + 2*x*y^3;
g[x_,y_]:= 1 + 3*x^3 - 2*y + 3*x^2*y^2;

F[x_,y_]:= x^2 y^3 +3x^3 y +x^3 - y^2 +y

Dt[F[x, y]]

Collect[%,{ Dt[x], Dt[y]}]

D[F[x,y],x]==f[x,y]

D[F[x,y],y]==g[x,y]

Solve[F[x,y]==c,{y}]

c

Remove["Global`*"];

f[x_,y_]:= 3*x^2 + 9*x^2*y + 2*x*y^3;
g[x_,y_]:= 1 + 3*x^3 - 2*y + 3*x^2*y^2;

Flatten[DSolve[{y'[x]==-f[x,y[x]]/g[x,y[x]],y[2]== -0.25},y[x],x ]];

solution=NDSolve[{y'[x]==-f[x,y[x]]/g[x,y[x]],y[2]== -0.25},y,{x,1.9,3},
    WorkingPrecision->24];Plot[y[x]/.solution,{x,1.9,3}];

d

Remove["Global`*"];

f[x_,y_]:= 3*x^2 + 9*x^2*y + 2*x*y^3;
g[x_,y_]:= 1 + 3*x^3 - 2*y + 3*x^2*y^2;

solv2 = DSolve[y'[x]==-(f[x,y[x]]+x)/g[x,y[x]],y[x],x ]

z[x_]:=y[x]/.solv2[[1]]

z[x]/.C[1]->c

Plot[Evaluate[Table[z[x]/.C[1]->c,{c,-5,5}]],{x,1,10}];

2

a) Plot der exakten Lösung

Remove["Global`*"];
DSolve[y''[x]- x y'[x]+ 2 y[x]==0,y,x]

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]- x y'[x]+ 2 y[x]==0, y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-5,5}];

b) Plot der exakten Lösung

Remove["Global`*"];
DSolve[y''[x]+ x y'[x]+ 2 y[x]==0,y,x]

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]+ x y'[x]+ 2 y[x]==0, y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-5,5}];

c) Plot der exakten Lösung

Remove["Global`*"];
DSolve[y''[x]+ x y'[x]+ 2 y[x]==x,y,x]

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]+ x y'[x]+ 2 y[x]==x, y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-5,5}];

d) Plot der exakten Lösung

Remove["Global`*"];
DSolve[y''[x]+ x y'[x]+ 2 y[x]==x^2,y,x]

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]+ x y'[x]+ 2 y[x]==x^2, y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-5,5}];

e) Plot der exakten Lösung

Remove["Global`*"];
DSolve[y''[x]+ y'[x]+ x^2 y[x]==0,y,x]

Remove["Global`*"];
solution=NDSolve[{y''[x]+ y'[x]+ x^2  y[x]==0, y[0]==1,y'[0]==1},y,{x,-5,5},
    WorkingPrecision->24];Plot[y[x]/.solution,{x,-5,5}];

f) Plot der exakten Lösung

Remove["Global`*"];
DSolve[y''[x]+ x y'[x]+ x^2 y[x]==0,y,x]

Remove["Global`*"];
solution=NDSolve[{y''[x]+ x y'[x]+ x^2  y[x]==0, y[0]==1,y'[0]==1},y,{x,-5,5},
    WorkingPrecision->24];Plot[y[x]/.solution,{x,-5,5},PlotRange->{-0.3,2}];

g) Plot der Lösung

Remove["Global`*"];
DSolve[y''[x]+ x^2 y'[x]+ x^2 y[x]==0,y,x]

Remove["Global`*"];
solution=NDSolve[{y''[x]+ x^2 y'[x]+ x^2  y[x]==0, y[0]==1,y'[0]==1},y,{x,-5,5},
    WorkingPrecision->24];Plot[y[x]/.solution,{x,-5,5},PlotRange->{-5,2}];

h) Plot der Lösung

Remove["Global`*"];
DSolve[y''[x]+ x y'[x]+ Cos[x] y[x]==0,y,x]

Remove["Global`*"];
solution=NDSolve[{y''[x]+ x y'[x]+ Cos[x] y[x]==0, y[0]==1,y'[0]==1},y,{x,-5,5},
    WorkingPrecision->24];Plot[y[x]/.solution,{x,-5,5},PlotRange->{-5,2}];

3

a

Remove["Global`*"];
DSolve[y''[x]+ a y'[x]+ b y[x]==0,y,x]

b

Remove["Global`*"];
solv = Flatten[
DSolve[{y'[x]+ y[x]==0, y[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-5,5}];

c

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]+ y'[x]+ y[x]==0, y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-5,5}];

d

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]- y'[x]+ y[x]==0, y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-5,7}];
Plot[y[x],{x,-10,2}];

e

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]+ y'[x]- y[x]==0, y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-5,7}];

f

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]+ y'[x]- y[x]==x, y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-5,7}];

g

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]+ y'[x]- y[x]==x^5, y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-3,3}];

h

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]+ y'[x]- y[x]==Cos[x], y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-7,7}];

i

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]- y'[x]- y[x]==Cos[x], y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-7,4}];

j

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]- y'[x]- y[x]==x+Cos[x/5+1], y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-7,4}];

k

Remove["Global`*"];
solv = Flatten[
DSolve[{y''[x]- y'[x]-2 y[x]==x+Cos[x/5+1], y[0]==1,y'[0]==1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-7,4}];

l

Remove["Global`*"];
solv = Flatten[
DSolve[{y'''[x]+y''[x]- y'[x]-2 y[x]==x, y[0]==1,y'[0]==1,y''[0]==-1},y,x]];
y = y/.solv;
Print[Simplify[y[x]]];
Plot[y[x],{x,-7,4}];

(* Das Programm findet keine klassische Darstellung der Lösung * )