gleichung[m_,d_,f_,funktion_]:={m*y''[x]+ d*y'[x]+ f*y[x] Š funktion, y[0]Š1, y'[0]Š0};

Lösungen

1 Schwingungen

Ÿ a

Remove["Global`*"];

gleichung[m_,d_,f_,funktion_]:={m*y''[x]+ d*y'[x]+ f*y[x] Š funktion, y[0]Š1, y'[0]Š0};

m=1; d=0; f=1; funktion=0;

solv =

DSolve[gleichung[m,d,f,funktion],y,x];

y = y/.solv[[1]];

Print["m = ",m," d = ",d," f = ",f," funktion = ",funktion];

Print["==> y[x] = ",y[x]//Simplify];

Plot[y[x],{x,0,4Pi}];

m = 1 d = 0 f = 1 funktion = 0

==> y @ x D = Cos @ x D

2 4 6 8 10 12

-1

-0.5

0.5

1

Ÿ b

Remove["Global`*"];

(* Neues f = 1/2 * altes f *)

gleichung[m_,d_,f_,funktion_]:={m*y''[x]+ d*y'[x]+ f*y[x] Š funktion, y[0]Š1, y'[0]Š0};

m=1; d=0; f=1/2; funktion=0;

solv =

DSolve[gleichung[m,d,f,funktion],y,x];

y = y/.solv[[1]];

Print["m = ",m," d = ",d," f = ",f," funktion = ",funktion];

Print["==> y[x] = ",y[x]//Simplify];

Plot[y[x],{x,0,4Pi}];

m = 1 d = 0 f = 1

€€€€ 2 funktion = 0

==> y @ x D = Cos A x

€€€€€€€€€€ •!!!! 2 E

2 4 6 8 10 12

-1

-0.5

0.5

1

Ÿ c

Remove["Global`*"];

gleichung[m_,d_,f_,funktion_]:={m*y''[x]+ d*y'[x]+ f*y[x] Š funktion, y[0]Š1, y'[0]Š0};

m=2; d=0; f=1; funktion=0;

solv =

DSolve[gleichung[m,d,f,funktion],y,x];

y = y/.solv[[1]];

Print["m = ",m," d = ",d," f = ",f," funktion = ",funktion];

Print["==> y[x] = ",y[x]//Simplify];

Plot[y[x],{x,0,4Pi}];

m = 2 d = 0 f = 1 funktion = 0

==> y @ x D = Cos A x

€€€€€€€€€€ •!!!! 2 E

2 4 6 8 10 12

-1

-0.5

0.5

1

Ÿ d

Remove["Global`*"];

gleichung[m_,d_,f_,funktion_]:={m*y''[x]+ d*y'[x]+ f*y[x] Š funktion, y[0]Š1, y'[0]Š0};

m=1; d=0; f=1; funktion=10;

solv =

DSolve[gleichung[m,d,f,funktion],y,x];

y = y/.solv[[1]];

Print["m = ",m," d = ",d," f = ",f," funktion = ",funktion];

Print["==> y[x] = ",y[x]//Simplify];

Plot[y[x],{x,0,4Pi}];

m = 1 d = 0 f = 1 funktion = 10

==> y @ x D = 10 - 9 Cos @ x D

2 4 6 8 10 12

2.5

5

7.5

10

12.5

15

17.5

Ÿ e

Remove["Global`*"];

gleichung[m_,d_,f_,funktion_]:={m*y''[x]+ d*y'[x]+ f*y[x] - f*(-y[x])Š funktion, y[0]Š1, y'[0]Š0};

m=1; d=0; f=1; funktion=0;

solv =

DSolve[gleichung[m,d,f,funktion],y,x];

y = y/.solv[[1]];

Print["m = ",m," d = ",d," f = ",f," funktion = ",funktion];

Print["==> y[x] = ",y[x]//Simplify];

Plot[y[x],{x,0,4Pi}];

m = 1 d = 0 f = 1 funktion = 0

==> y @ x D = Cos @•!!!! 2 x D

2 4 6 8 10 12

-1 -0.5 0.5 1

Ÿ f

Remove["Global`*"];

links1 = 1*y1''[x] + 1*y1[x] - 1*(y2[x]-y1[x]);

links2 = 1*y2''[x] + 1*y2[x] - 1*(y1[x]-y2[x]);

{links1 + links2 == 0+0, links1 - links2 == 0+0}

8 y1 @ x D + y2 @ x D + y1

@ x D + y2

@ x D Š 0, 3 y1 @ x D - 3 y2 @ x D + y1

@ x D - y2

@ x D Š 0 <

Sei r[x_] := y1[x] + y2[x], s[x_] := y1[x] - y2[x]

links1 + links2

y1 @ x D + y2 @ x D + y1

@ x D + y2

@ x D

u1 = links1 + links2 /. {y1[x]+y2[x] -> r[x]} /.{y1''[x]+y2''[x] -> r''[x]}

r @ x D + r

@ x D links1 - links2

3 y1 @ x D - 3 y2 @ x D + y1

@ x D - y2

@ x D u2 = links1 - links2

3 y1 @ x D - 3 y2 @ x D + y1

@ x D - y2

@ x D

u2 = links1 - links2 /. {3 y1[x]- 3 y2[x] -> 3 s[x]} /.{y1''[x]-y2''[x] -> s''[x]}

3 s @ x D + s

@ x D

Das ergibt die Differentialgleichungen:

r[x] + r''[x] = 0, r[x] = y1[x] + y2[x], r[0] = 6, r'[0] = 0 und s[x] + s''[x] = 0, s[x] = y1[x] - y2[x], s[0] = -4, s'[0] = 0 Diese Gleichungen sind unabhängig lösbar.

Remove[m,d,f,funktion,r,s,x];

gleichung[m_,d_,f_,funktion_]:={m*r''[x]+ d*r'[x]+ f*r[x] Š funktion, r[0]Š3, r'[0]Š0};

m=1; d=0; f=1; funktion=0;

solv =

DSolve[gleichung[m,d,f,funktion],r,x];

r = r/.solv[[1]];

Print["m = ",m," d = ",d," f = ",f," funktion = ",funktion];

Print["==> r[x] = ",r[x]//Simplify];

Plot[r[x],{x,0,4Pi}];

m = 1 d = 0 f = 1 funktion = 0

==> r @ x D = 3 Cos @ x D

2 4 6 8 10 12

-1

1

2

3

Remove[m,d,f,funktion,s,x];

gleichung[m_,d_,f_,funktion_]:={m*s''[x]+ d*s'[x]+ f*s[x] Š funktion, s[0]Š1, s'[0]Š0};

m=1; d=0; f=3; funktion=0;

solv =

DSolve[gleichung[m,d,f,funktion],s,x];

s = s/.solv[[1]];

Print["m = ",m," d = ",d," f = ",f," funktion = ",funktion];

Print["==> s[x] = ",s[x]//Simplify];

Plot[s[x],{x,0,4Pi}];

m = 1 d = 0 f = 3 funktion = 0

==> s @ x D = Cos @•!!!! 3 x D

2 4 6 8 10 12

-1 -0.5 0.5 1

y1[x_]:= (r[x]+s[x])/2;

y2[x_]:= (r[x]-s[x])/2 -6; (* Koordinatenverschiebung -6 *) Print["==> y1[x] = ",y1[x]//Simplify];

Plot[y1[x],{x,0,4Pi}];

Print["==> y2[x] = ",y2[x]-6//Simplify];

Plot[y2[x],{x,0,4Pi}];

Plot[{y1[x],y2[x],-6},{x,0,4Pi}];

==> y1 @ x D = 1

€€€€ 2 H 3 Cos @ x D + Cos @•!!!! 3 x DL

2 4 6 8 10 12

-2 -1 1 2

==> y2 @ x D = 1

€€€€ 2 H -24 + 3 Cos @ x D - Cos @•!!!! 3 x DL

2 4 6 8 10 12 -7

-6 -5 -4

2 4 6 8 10 12

-8 -6 -4 -2 2

Plot[{y1[x],y2[x],-6},{x,0,12Pi}];

5 10 15 20 25 30 35

-4

-2

2

Ÿ g

Remove["Global`*"];

gleichung[m_,d_,f_,funktion_]:={m*y''[x]+ d*y'[x]+ f*y[x] Š funktion, y[0]Š1, y'[0]Š0};

m=1; d=0; f=1; funktion=Sin[x];

solv =

DSolve[gleichung[m,d,f,funktion],y,x];

y = y/.solv[[1]];

Print["m = ",m," d = ",d," f = ",f," funktion = ",funktion];

Print["==> y[x] = ",y[x]//Simplify];

Plot[y[x],{x,0,4Pi}];

m = 1 d = 0 f = 1 funktion = Sin @ x D

==> y @ x D = 1

€€€€ 2 H - H -2 + x L Cos @ x D + Sin @ x DL

2 4 6 8 10 12

-4 -2 2

Plot[y[x],{x,0,20Pi}];

10 20 30 40 50 60

-30

-20

-10

10

20

30

Ÿ h

Remove["Global`*"];

gleichung[m_,d_,f_,funktion_]:={m*y''[x]+ d*y'[x]+ f*y[x] Š funktion, y[0]Š1, y'[0]Š0};

m=1; d=1; f=1; funktion= 0;

solv =

DSolve[gleichung[m,d,f,funktion],y,x];

y = y/.solv[[1]];

Print["m = ",m," d = ",d," f = ",f," funktion = ",funktion];

Print["==> y[x] = ",y[x]//Simplify];

Plot[y[x],{x,0,4Pi}];

m = 1 d = 1 f = 1 funktion = 0

==> y @ x D = 1

€€€€ 3 ã

i

k jjj 3 Cos A •!!!! 3 x

€€€€€€€€€€€€€€ 2 E + •!!!! 3 Sin A •!!!! 3 x

€€€€€€€€€€€€€€ 2 Ey { zzz

2 4 6 8 10 12

-0.1

0.1

0.2

0.3

Ÿ i

Remove["Global`*"];

gleichung[m_,d_,f_,funktion_]:={m*y''[x]+ d*y'[x]+ f*y[x] Š funktion, y[0]Š1, y'[0]Š0};

m=1; d=1; f=1; funktion= Sin[x];

solv =

DSolve[gleichung[m,d,f,funktion],y,x];

y = y/.solv[[1]];

Print["m = ",m," d = ",d," f = ",f," funktion = ",funktion];

Print["==> y[x] = ",y[x]//Simplify];

Plot[y[x],{x,0,4Pi}];

m = 1 d = 1 f = 1 funktion = Sin @ x D

==> y @ x D = - Cos @ x D + 2

€€€€ 3 ã

i

k jjj 3 Cos A •!!!! 3 x

€€€€€€€€€€€€€€ 2 E + •!!!! 3 Sin A •!!!! 3 x

€€€€€€€€€€€€€€ 2 Ey { zzz

2 4 6 8 10 12

-1 -0.5 0.5 1

Plot[y[x],{x,0,12Pi}];

5 10 15 20 25 30 35

-1 -0.5 0.5 1

Ÿ j

Remove["Global`*"];

f1 = 3; f2 = 4;

geichungen= {F == f x, F == a x1, F == b x2, x == x1+x2};

solv = Solve[geichungen, {f,x1,x2,x,F}]

Solve::svars : Equations may not give solutions for all "solve" variables.

Mehr…

98 x1 ® 0, x2 ® 0, F ® 0, x ® 0 < , 9 f ® a b

€€€€€€€€€€€€

a + b , x1 ® b x

€€€€€€€€€€€€

a + b , x2 ® a x

€€€€€€€€€€€€

a + b , F ® a b x

€€€€€€€€€€€€€

a + b ==

solv[[2]][[1]]

f ® a b

€€€€€€€€€€€€

a + b

solv1=solv[[2]][[1]] /. {a->3, b->4}

f ® 12

€€€€€€€

7

f = f /. solv1

€€€€€€€ 12 7

gleichung[m_,d_,f_,funktion_]:={m*y''[x]+ d*y'[x]+ f*y[x] Š funktion, y[0]Š0, y'[0]Š1};

m=2; d=1; f=f; funktion= 4 Sin[2 x];

solv =

DSolve[gleichung[m,d,f,funktion],y,x];

y = y/.solv[[1]];

Print["m = ",m," d = ",d," f = ",f," funktion = ",funktion];

Print["==> y[x] = ",y[x]//Simplify];

Plot[y[x],{x,0,4Pi}];

m = 2 d = 1 f = 12

€€€€€€€ 7 funktion = 4 Sin @ 2 x D

==> y @ x D = 1

€€€€€€€€€€€€€€€€

47437 i

k jjjjjj 2 i

k jjjjjj -4361 Cos @ 2 x D + 4361 ã

Cos A 1

€€€€ 4 $%%%%%%%%%% 89

€€€€€€€

7 x E - 13706 Sin @ 2 x D + 2347 •!!!!!!!!!! 623 ã

Sin A 1

€€€€ 4 $%%%%%%%%%% 89

€€€€€€€

7 x E y { zzzzzz y

{ zzzzzz

2 4 6 8 10 12

-1 -0.5 0.5 1 1.5 2

Plot[y[x],{x,0,12Pi}];

0.5

1

1.5

2

2 Linienintegrale

Ÿ a

Ÿ i

Remove["Global`*"];

F[x_,y_,z_]:= {3 x y, 5 z, 10 x};

v[t]:= {t^2+1,2 t^2,t^3};

D[v[t],t]

8 2 t, 4 t, 3 t

<

integrand[t_]:= F[x,y,z].Evaluate[D[v[t],t]] /.{x->v[t][[1]], y->v[t][[2]], z->v[t][[3]]};

integrand[t]

20 t

+ 30 t

H 1 + t

L + 12 t

H 1 + t

L integrand[t]/.t->u

20 u

+ 30 u

H 1 + u

L + 12 u

H 1 + u

L

Integrate[Evaluate[integrand[t]/.t->u],{u,0,2}]

576

Ÿ ii

Remove["Global`*"];

F[x_,y_,z_]:= {3 x y, 5 z, 10 x};

absF[x_,y_,z_]:= Sqrt[F[x,y,z].F[x,y,z]];

absF[x,y,z]

•!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 100 x

+ 9 x

y

+ 25 z

v @ t D := 8 t ^ 2 + 1, 2 t ^ 2, t ^ 3 < ; Sqrt @ D @ v @ t D , t D .D @ v @ t D , t DD

•!!!!!!!!!!!!!!!!!!!!!!!!!!! 20 t

+ 9 t

integrand[t_]:= absF[x,y,z]* Evaluate[Sqrt[D[v[t],t].D[v[t],t]]] /.{x->v[t][[1]], y->v[t][[2]], z->v[t][[3]]};

integrand[t]

•!!!!!!!!!!!!!!!!!!!!!!!!!!! 20 t

+ 9 t

"####################################################################################### 25 t

+ 100 H 1 + t

L

+ 36 t

H 1 + t

L

integrand[t]/.t->u

•!!!!!!!!!!!!!!!!!!!!!!!!!!! 20 u

+ 9 u

"####################################################################################### 25 u

+ 100 H 1 + u

L

+ 36 u

H 1 + u

L

Integrate[Evaluate[integrand[t]/.t->u],{u,0,2}]

à

2

•!!!!!!!!!!!!!!!!!!!!!!!!!!! 20 u

+ 9 u

"####################################################################################### 25 u

+ 100 H 1 + u

L

+ 36 u

H 1 + u

L

âu

NIntegrate[Evaluate[integrand[t]/.t->u],{u,0,2}]

758.233

Ÿ b

Ÿ i

Remove["Global`*"];

F[x_,y_,z_]:= {3 x y, 5 z, 10 x};

v[t]:= {t^3,2 t^2,t^2+1};

D[v[t],t]

8 3 t

, 4 t, 2 t <

integrand[t_]:= F[x,y,z].Evaluate[D[v[t],t]] /.{x->v[t][[1]], y->v[t][[2]], z->v[t][[3]]};

integrand[t]

20 t

+ 18 t

+ 20 t H 1 + t

L integrand[t]/.t->u

20 u

+ 18 u

+ 20 u H 1 + u

L

Integrate[Evaluate[integrand[t]/.t->u],{u,0,2}]

824

Ÿ ii

Remove["Global`*"];

F[x_,y_,z_]:= {3 x y, 5 z, 10 x};

absF[x_,y_,z_]:= Sqrt[F[x,y,z].F[x,y,z]];

absF[x,y,z]

•!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 100 x

+ 9 x

y

+ 25 z

8 2 t, 4 t, 3 t

<

v @ t D := 8 t ^ 3, 2 t ^ 2, t ^ 2 + 1 < ; Sqrt @ D @ v @ t D , t D .D @ v @ t D , t DD

•!!!!!!!!!!!!!!!!!!!!!!!!!!! 20 t

+ 9 t

integrand[t_]:= absF[x,y,z]* Evaluate[Sqrt[D[v[t],t].D[v[t],t]]] /.{x->v[t][[1]],

integrand[t]/.t->u

•!!!!!!!!!!!!!!!!!!!!!!!!!!! 20 u

+ 9 u

"################################################################### 100 u

+ 36 u

+ 25 H 1 + u

L

Integrate[Evaluate[integrand[t]/.t->u],{u,0,2}]

à

0

2

•!!!!!!!!!!!!!!!!!!!!!!!!!!! 20 u

+ 9 u

"################################################################### 100 u

+ 36 u

+ 25 H 1 + u

L

âu

NIntegrate[Evaluate[integrand[t]/.t->u],{u,0,2}]

913.579

Ÿ c

Ÿ i

Remove["Global`*"];

F[x_,y_,z_]:= {3 x y, 5 z, 10 x};

v[t]:={Cos[t],Sin[2 t],1+Sin[3 t]};

D[v[t],t]

8-Sin @ t D , 2 Cos @ 2 t D , 3 Cos @ 3 t D<

integrand[t_]:= F[x,y,z].Evaluate[D[v[t],t]] /.{x->v[t][[1]], y->v[t][[2]], z->v[t][[3]]};

integrand[t]

30 Cos @ t D Cos @ 3 t D - 3 Cos @ t D Sin @ t D Sin @ 2 t D + 10 Cos @ 2 t D H 1 + Sin @ 3 t DL integrand[t]/.t->u

30 Cos @ u D Cos @ 3 u D - 3 Cos @ u D Sin @ u D Sin @ 2 u D + 10 Cos @ 2 u D H 1 + Sin @ 3 u DL (* Integrate[Evaluate[integrand[t]/.t->u],{u,0,2Pi}] *)

NIntegrate[Evaluate[integrand[t]/.t->u],{u,0,2Pi}]

-4.71239

Ÿ ii

Remove["Global`*"];

F[x_,y_,z_]:= {3 x y, 5 z, 10 x};

absF[x_,y_,z_]:= Sqrt[F[x,y,z].F[x,y,z]];

absF[x,y,z]

•!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 100 x

+ 9 x

y

+ 25 z

8 2 t, 4 t, 3 t

<

v @ t D := 8 Cos @ t D , Sin @ 2 t D , 1 + Sin @ 3 t D< ; Sqrt @ D @ v @ t D , t D .D @ v @ t D , t DD

"#################################################################################### 4 Cos @ 2 t D

+ 9 Cos @ 3 t D

+ Sin @ t D

integrand[t_]:= absF[x,y,z]* Evaluate[Sqrt[D[v[t],t].D[v[t],t]]] /.{x->v[t][[1]], y->v[t][[2]], z->v[t][[3]]};

integrand[t]

"#################################################################################### 4 Cos @ 2 t D

+ 9 Cos @ 3 t D

+ Sin @ t D

"################################################################################################################################# 100 Cos @ t D

+ 9 Cos @ t D

Sin @ 2 t D

+ 25 H 1 + Sin @ 3 t DL

integrand[t]/.t->u

"#################################################################################### 4 Cos @ 2 u D

+ 9 Cos @ 3 u D

+ Sin @ u D

"################################################################################################################################# 100 Cos @ u D

+ 9 Cos @ u D

Sin @ 2 u D

+ 25 H 1 + Sin @ 3 u DL

(* Integrate[Evaluate[integrand[t]/.t->u],{u,0,2}] *) NIntegrate[Evaluate[integrand[t]/.t->u],{u,0,2 Pi}]

138.451

3 Allgemeine Lösungen wichtiger Differentialgleichungen mit der Maschine

Ÿ a) y'(x) + a y(x) = f(x)

Remove @ "Global`*" D ;

solv1 = DSolve @ y ' @ x D + a y @ x D Š f @ x D , y, x D ; u @ x_ D := y • . solv1 @@ 1 DD ;

a = H u @ x D@ x D •• InputForm L@@ 1 DD@@ 2 DD@@ 2 DD@@ 2 DD@@ 1 DD ; v @ x_ D := u @ x D@ x D • . a ® t;

Print @ "y H x L = ", v @ x DD ;

Print @ " = ", v @ x D •• Simplify D ; v @ w D •• Simplify

y H x L = ã

C @ 1 D + ã

à

1 x

ã

f @ t D â t

-xa

i jj @ D

@ D â y zz

Ÿ b) y''(x) + a y'(x) + b y(x) = f(x)

Remove @ "Global`*" D ;

solv2 = DSolve @ y '' @ x D + a y ' @ x D + b y @ x D Š f @ x D , y, x D@@ 1 DD ; u @ x_ D := y • . solv2 @@ 1 DD ;

a = u @ x D@@ 2 DD@@ 3 DD@@ 2 DD@@ 2 DD@@ 1 DD ; b = u @ x D@@ 2 DD@@ 4 DD@@ 2 DD@@ 2 DD@@ 1 DD ; v @ x_ D := u @ x D@ x D • . 8 a ® t, b ® t < ; Print @ "y H x L = ", v @ x DD ;

Print @ " = ", v @ x D •• Simplify D ; v @ w D •• Simplify

y H x L = ã

C @ 1 D + ã

C @ 2 D + ã

à

1

x

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ ã

•!!!!!!!!!!!!!!!!!!

a

-

€€€€€€€€€€€€€€€€ 4

b

f €€€€€€€€€€ @ t D ât + ã

à

1 x

- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ ã

•!!!!!!!!!!!!!!!!!!

a

-

€€€€€€€€€€€€€€€€ 4

b

f €€€€€€€€€€ @ t D ât

=

ã

i

k jjjj C @ 1 D + ã

C @ 2 D + ã

à

x

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ ã

•!!!!!!!!!!!!!!!!!! a

-

4

€€€€€€€€€€€€€€€€€ b

f @ t D â t + à

1 x

- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ ã

•!!!!!!!!!!!!!!!!!! a

-

4

€€€€€€€€€€€€€€€€€ b

f @ t D â t y { zzzz ã

i

k jjjjj C @ 1 D + ã

C @ 2 D + ã

à

w

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ ã

•!!!!!!!!!!!!!!!!!! a

-

4

b

€€€€€€€€€€€€€€€€ f @ t D â t + à

1 w

- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ ã

•!!!!!!!!!!!!!!!!!! a

-

4

b

€€€€€€€€€€€€€€€€ f @ t D ât y

{ zzzzz

Ÿ c) Wahl von Parametern a = 2, b = -1, C[1] = 1, C[2] = 1, f(x) = cos(x), Plot

Ÿ Maschinenlösung

Remove @ "Global`*" D ;

solv2 = DSolve @ y '' @ x D + a y ' @ x D + b y @ x D Š f @ x D , y, x D@@ 1 DD ; u @ x_ D := y • . solv2 @@ 1 DD ;

a = u @ x D@@ 2 DD@@ 3 DD@@ 2 DD@@ 2 DD@@ 1 DD ; b = u @ x D@@ 2 DD@@ 4 DD@@ 2 DD@@ 2 DD@@ 1 DD ; v @ x_ D := u @ x D@ x D • . 8 a ® t, b ® t < ;

s @ u_ D := v @ x D • . 8 x ® u, a ® 2, b ® -1, C @ 1 D ® 1, C @ 2 D ® 1, f @ t D ® Cos @ t D< ; Print @ "y H x L = ", s @ x D •• N •• Simplify D ;

Print @ "y H 0 L = ", s @ 0 D •• N D ;

Print @ "y' H 0 L = ", H D @ s @ m D , m D • . m ® 0 L •• N D ; Plot @ s @ m D , 8 m, 0, Pi < , AspectRatio ® 1 D ;

y H x L = 2.24228 ã

+ 0.876817 ã

- 0.25 Cos @ x D + 0.25 Sin @ x D y H 0 L = 2.86909

y' H 0 L = -4.80015

0.5 1 1.5 2 2.5 3

2.5 3 3.5

Ÿ Direkter Plot der exakten Lösung mit Ausgabe der Funktion

Remove["Global`*"];

solv = Flatten[DSolve[{y''[x]+2 y'[x]-1 y[x]Š Cos[x],y[0]==2.86909,y'[0]==-4.80015},y,x]];

y = y/.solv;

Print["y(x) = ",Simplify[y[x]]];

Print["y(0) = ", y[0], " y'(0) = ",(D[y[x],x] /. x->0) //N];

Plot[y[x],{x,0,Pi},AspectRatio®1];

Plot[y[x],{x,0,4 Pi},AspectRatio®1];

0.5 1 1.5 2 2.5 3 2.5

3 3.5

2 4 6 8 10 12

10 20 30 40 50

Ÿ Direkter Plot der numerischen Lösung

Remove["Global`*"];

solution=NDSolve[{y''[x]+2 y'[x]-1 y[x]Š

Cos[x],y[0]==2.86909,y'[0]==-4.80015},y,{x,0,4 Pi}];

Plot[y[x]/. solution,{x,0,Pi},AspectRatio®1];

Plot[y[x]/. solution,{x,0,4 Pi},AspectRatio®1];

0.5 1 1.5 2 2.5 3 2.5

3 3.5

10

20

30

40

50

Ÿ d) Direkter Plot der exakten Lösung einer anderen D'Gl.

Ÿ Maschinenlösung

Remove["Global`*"];

solv = Flatten[DSolve[{y''[x] + y'[x] - y[x] ŠCos[x+1], y[0]==1, y'[0]==0},y,x]];

y = y/.solv;

Print["y(x) = ",Simplify[y[x]]];

Print["Numerisch y(x) = ",Simplify[y[x]]//N];

Plot[y[x],{x,0,Pi},AspectRatio®1];

y H x L =

- 1

€€€€€€€ 10 ã

I -5 + •!!!! 5 - 5 ã

- •!!!! 5 ã

- 2 Cos @ 1 D - 2 ã

Cos @ 1 D + 4 ã

Cos @ 1 + x D + Sin @ 1 D -

•!!!! 5 Sin @ 1 D + ã

Sin @ 1 D + •!!!! 5 ã

Sin @ 1 D - 2 ã

Sin @ 1 + x DM

Numerisch y H x L = - 0.1 2.71828

H- 4.88465 - 5.59362 2.71828

+ 4. 2.71828

Cos @ 1. + x D - 2. 2.71828

Sin @ 1. + x DL

0.5 1 1.5 2 2.5 3

1.5

2

2.5

3

3.5

4

p1=Plot[y[x],{x,0,Pi},AspectRatio®1];

0.5 1 1.5 2 2.5 3

1.5 2 2.5 3 3.5 4

Ÿ Handlösung

solv=Solve[l^2+l-1==0,{l}][[1]];

loes1=l /. solv;

loes1//TeXForm

\frac{1}{2} \left(-1-\sqrt{5}\right) solv=Solve[l^2+l-1==0,{l}][[2]];

loes2=l /. solv;

loes2//TeXForm

\frac{1}{2} \left(-1+\sqrt{5}\right) Remove[x,y,c1,c2]

solv = Flatten[DSolve[y''[x] + y'[x] - y[x]==0,y,x]];

y = y/.solv;

y[x]

ã

•!!!!!5

€€€€€€€€€₂ Nx

C @ 1 D + ã

•!!!!!5

€€€€€€€€€₂ Nx

C @ 2 D Cos[x+1]//TrigExpand

Cos @ 1 D Cos @ x D - Sin @ 1 D Sin @ x D

solv3=Flatten[Simplify[Solve[{-2 r+s==Cos[1], -r-2s==-Sin[1]},{r,s}]]]

9 r ® 1

€€€€ 5 H-2 Cos @ 1 D + Sin @ 1 DL , s ® 1

€€€€ 5 H Cos @ 1 D + 2 Sin @ 1 DL=

Remove[y];

y[x_,c1_,c2_]:= c1 E^(1/2(-1-Sqrt[5])x)+c2 E^(1/2(-1+Sqrt[5])x)+ r Cos[x]+ s Sin[x];

y[x,c1,c2]//Simplify

€€€€ 1

5 I 5 ã

I c1 + c2 ã

M - 2 Cos @ 1 + x D + Sin @ 1 + x DM y[x,c1,c2]

c1 ã

+ c2 ã

+ 1

€€€€ 5 Cos @ x D H-2 Cos @ 1 D + Sin @ 1 DL + 1

€€€€ 5 H Cos @ 1 D + 2 Sin @ 1 DL Sin @ x D D[y[x,c1,c2],x]//Simplify

€€€€€€€ 1

10 I 2 Cos @ 1 + x D + ã

I-5 II 1 + •!!!! 5 M c1 - I-1 + •!!!! 5 M c2 ã

M + 4 ã

Sin @ 1 + x DMM y[0,c1,c2]

c1 + c2 + 1

€€€€ 5 H-2 Cos @ 1 D + Sin @ 1 DL (Simplify[D[y[x,c1,c2],x]]/.x->0)

€€€€€€€ 1

10 I-5 II 1 + •!!!! 5 M c1 - I-1 + •!!!! 5 M c2 M + 2 Cos @ 1 D + 4 Sin @ 1 DM

solv=Flatten[Solve[Evaluate[{y[0,c1,c2]==1, (D[y[x,c1,c2],x]/.x->0)==0}],{c1,c2} ]]

9 c1 ® 1

€€€€€€€

10 I 5 - •!!!! 5 + 2 Cos @ 1 D - Sin @ 1 D + •!!!! 5 Sin @ 1 DM , c2 ® €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 5 + 5 •!!!! 5 + 2 •!!!! €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 5 Cos @ 1 D - 5 Sin €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ @ 1 D - •!!!! 5 Sin €€€€€€€€€€€€€€€€€€€€ @ 1 D

10 •!!!! 5 =

c1=c1/.solv[[1]]//Simplify

€€€€€€€ 1

10 I 5 - •!!!! 5 + 2 Cos @ 1 D + I-1 + •!!!! 5 M Sin @ 1 DM c2=c2/.solv[[2]]//Simplify

€€€€€€€ 1

10 I 5 + •!!!! 5 + 2 Cos @ 1 D - I 1 + •!!!! 5 M Sin @ 1 DM y[x,c1,c2]

€€€€ 1

5 Cos @ x D H-2 Cos @ 1 D + Sin @ 1 DL + 1

€€€€€€€

10 ã

I 5 - •!!!! 5 + 2 Cos @ 1 D + I-1 + •!!!! 5 M Sin @ 1 DM +

€€€€€€€ 1

10 ã

I 5 + •!!!! 5 + 2 Cos @ 1 D - I 1 + •!!!! 5 M Sin @ 1 DM + 1

€€€€ 5 H Cos @ 1 D + 2 Sin @ 1 DL Sin @ x D y[x,c1,c2]//Simplify

€€€€€€€ 1

10 I 2 Cos @ x D H-2 Cos @ 1 D + Sin @ 1 DL + ã

I 5 - •!!!! 5 + 2 Cos @ 1 D + I-1 + •!!!! 5 M Sin @ 1 DM +

ã

I 5 + •!!!! 5 + 2 Cos @ 1 D - I 1 + •!!!! 5 M Sin @ 1 DM + 2 H Cos @ 1 D + 2 Sin @ 1 DL Sin @ x DM

y[x,c1,c2]//N//Simplify

0.488465 ã

+ 0.559362 ã

- 0.0478267 Cos @ x D + 0.444649 Sin @ x D

p2 = Plot[y[x,c1,c2],{x,0,Pi},AspectRatio®1];

0.5 1 1.5 2 2.5 3

1.5 2 2.5 3 3.5 4

Show[p1,p2];

2

2.5

3

3.5

4

Remove[x,y];

solv = Flatten[DSolve[{y''[x] + y'[x] - y[x] ŠCos[x+1], y[0]==1, y'[0]==0},y,x]];

y = y/.solv;

Print["y(x) = ",Simplify[y[x]]];

Print["Numerisch y(x) = ",Simplify[y[x]]//N];

Plot[y[x],{x,0,Pi},AspectRatio®1];

y H x L =

- 1

€€€€€€€ 10 ã

I- 5 + •!!!! 5 - 5 ã

- •!!!! 5 ã

- 2 Cos @ 1 D - 2 ã

Cos @ 1 D + 4 ã

Cos @ 1 + x D + Sin @ 1 D -

•!!!! 5 Sin @ 1 D + ã

Sin @ 1 D + •!!!! 5 ã

Sin @ 1 D - 2 ã

Sin @ 1 + x DM

Numerisch y H x L = - 0.1 2.71828

H- 4.88465 - 5.59362 2.71828

+ 4. 2.71828

Cos @ 1. + x D - 2. 2.71828

Sin @ 1. + x DL

0.5 1 1.5 2 2.5 3

1.5 2 2.5 3 3.5 4

y[x]//Simplify//N

-0.1 2.71828

H-4.88465 - 5.59362 2.71828

+

4. 2.71828

Cos @ 1. + x D - 2. 2.71828

Sin @ 1. + x DL