第三章 拉普拉斯s微分方程求解

第三章 程序

例3-1............................................................................................................................... 1

例3-2............................................................................................................................... 2

例3-3............................................................................................................................... 3

例3-4............................................................................................................................... 3

例3-5............................................................................................................................... 4

例3-6............................................................................................................................... 4

例3-7............................................................................................................................... 6

例3-9............................................................................................................................... 7

例3-10 . ............................................................................................................................ 7

例3-11 . ............................................................................................................................ 8

例3-12 . ............................................................................................................................ 8

例3-13 . ............................................................................................................................ 9

例3-14 . ............................................................................................................................ 9

例3-15 . .......................................................................................................................... 10

例3-16 . ...........................................................................................................................11

例3-17 . .......................................................................................................................... 12

例3-18 . .......................................................................................................................... 12

例3-19 . .......................................................................................................................... 13

例3-20 . .......................................................................................................................... 14

例3-21 . .......................................................................................................................... 14

例3-22 . .......................................................................................................................... 15

例3-23 . .......................................................................................................................... 15

例3-24 . .......................................................................................................................... 15

例3-1

>> A=[1 3 4]

A =

1 3 4

>> B=[1;3;4]

B =

1

3

4

>> C=[1 2 3;4 5 6]

C =

1 2 3

4 5 6

>> D=4.5

D =

4.5000

>> E=[]

E =

[]

>> C=[1 2 3;4 5]

??? Error using ==> vertcat

CA T arguments dimensions are not consistent.

例3-2

>> a=1:5

a =

1 2 3 4 5

>> t=0:pi/4:pi

t =

Columns 1 through 4

0 0.7854 1.5708 2.3562

Column 5

3.1416

>> b=pi:-pi/4:0

b =

3.1416 2.3562 1.5708 0.7854

>> CC=[C' C']

CC =

1 4 1 4

2 5 2 5

3 6 3 6

>> C(3,4)=10

C =

1 2 3 0

4 5 6 0

0 0 0 10

例3-3

>> yourName=input('请输入您的姓名：\n','s')

请输入您的姓名：

zhao

yourName =

zhao

>> yourAge=input('请输入您的年龄：')

请输入您的年龄：33

yourAge =

33

例3-4

>> A=magic(3)

A =

8 1 6

3 5 7

4 9 2

>> B=eye(size(A))

B =

1 0 0

0 1 0

0 0 1

>> t=logspace(0,5,6)

t =

Columns 1 through 4

1 10

Columns 5 through 6

10000 100000

例3-5

>> load -ascii mydata.dat

>> mydata

mydata =

16.0000 2.0000 3.0000

5.0000 11.0000 10.0000

9.0000 7.0000 6.0000

4.0000 14.0000 15.0000

-5.7000 -5.7000 -5.7000

-5.7000 -5.7000 -5.7000

8.0000 6.0000 4.0000

例3-6

>> A=magic(3)

A =

8 1 6

3 5 7

4 9 2

>> a22=A(2,2)

a22 = 100 13.0000 8.0000 12.0000 1.0000 -5.7000 -5.7000 2.0000 1000

5

>> A1=A(1:2:3,[2,3])

A1 =

1 6

9 2

>> A2=A(:,2:end)

A2 =

1 6

5 7

9 2

>> L1=logical([1 0 1]);

>> L2=logical([1 1 0]);

>> A3=A(L1,L2)

A3 =

8 1

4 9

>> A4=A;

>> A4(1,2)=0

A4 =

8 0 6

3 5 7

4 9 2

>> A4(4,4)=121

A4 =

8 0 6

3 5 7

4 9 2

0 0 0 0 0 0 121

>> A5=A;

>> A5(:,3)=[]

A5 =

8 1

3 5

4 9

>> A6=rot90(A)

A6 =

6 7 2

1 5 9

8 3 4

例3-7

>> clear

>> a = [2 4 6 5]; b = ones(4, 4) * 3; c =magic(4);

>> save -ascii mydat.dat

>> clear

>> load -ascii mydat.dat

>> mydat

mydat =

2 4 6 5

3 3 3 3

3 3 3 3

3 3 3 3

3 3 3 3

16 2 3 13

5 11 10 8

9 7 6 12

4 14 15 1

>> clear

>> a = [2 4 6 5]; b = ones(4, 4) * 3; c =magic(4);

>> save mydat

>> clear

>> load mydat

>> whos

Name Size Bytes Class Attributes

a 1x4 32 double b 4x4 128 double c 4x4 128 double

例3-9

>> A=magic(3)

A =

8 1

3 5

4 9

>> B=eye(3)

B =

1 0

0 1

0 0

>> C=A+B

C =

9 1

3 6

4 9

例3-10

>> D=A*C

D =

99 68

70 96

71 76

>> D1=A.*C

D1 =

6 7 2 0 0 1 6 7 3 73 74 93

72 1 36

9 30 49

16 81 6

>> D2=3*magic(3)

D2 =

24 3 18

9 15 21

12 27 6

例3-11

>> A =[8 1 6; 3 5 7; 4 9 2];

>> B=[1 2 3]';

>> X=A\B

X =

0.0500

0.3000

0.0500

例3-12

>> A=magic(3);

>> B=2;

>> C=A^B

C =

91 67 67

67 91 67

67 67 91

>> B^A

ans =

1.0e+004 *

1.0942 1.0906 1.0921

1.0912 1.0933 1.0924

1.0915 1.0930 1.0923

例3-13

>> A = [1 1 1;1 2 3;1 3 6]

A =

1 1 1

1 2 3

1 3 6

>> X = A.^2

X =

1 1

1 4

1 9

>> X1=2.^A

X1 =

2 2

2 4

2 8

>> C=X.^B

C =

1

1

1

例3-14

>> A=magic(3)

A =

8 1

3 5

4 9

>> A'

1 9 36 2 8 64 1 16 81 6 7 2 1 81 1296

ans =

8 3 4

1 5 9

6 7 2

>> B=i*eye(3)

B =

0 + 1.0000i 0 0 0 0 + 1.0000i 0 0 0 0 + 1.0000i

>> C=(A+B)'

C =

8.0000 - 1.0000i 3.0000 4.0000 1.0000 5.0000 - 1.0000i 9.0000 6.0000 7.0000 2.0000 - 1.0000i

>> C=(A+B).'

C =

8.0000 + 1.0000i 3.0000 4.0000 1.0000 5.0000 + 1.0000i 9.0000 6.0000 7.0000 2.0000 + 1.0000i

例3-15

>> A=magic(3)

A =

8 1 6

3 5 7

4 9 2

>> B=[3 4 5;6 9 7;1 8 2]

B =

3 4 5

6 9 7

1 8 2

>> A>=B

ans =

1 0 1

0 0 1

1 1 1

>> A~=B

ans =

1 1 1

1 1 0

1 1 0

例3-16

>> A=[1 0 2;3 0 5]

A =

1 0 2

3 0 5

>> B=[2 6 1; 0 0 8]

B =

2 6 1

0 0 8

>> A&B

ans =

1 0 1

0 0 1

>> A|B

ans =

1 1 1

1 0 1

>> ~A

ans =

0 1 0

0 1 0

>> xor(A,B)

ans =

0 1 0

1 0 0

例3-17

>> str=input('input a string:\n','s');

if ~isempty(str) && ischar(str)

sprintf('Input string is ''%s''', str)

end

input a string:

MA TLAB is great

ans =

Input string is 'MATLAB is great'

例3-18

>> clear

>> f=sym('a*x^2+b*x+c')

f =

a*x^2+b*x+c

>> f-a

??? Undefined function or variable 'a'.

>> syms a b c x

>> f1=a*x^2+b*x+c

f1 =

a*x^2+b*x+c

>> f1-a

ans =

a*x^2+b*x+c-a

>> clear

>> A=sym('[a b;c d]')

A =

[ a, b]

[ c, d]

>> C=[a b;c d]

??? Undefined function or variable 'a'.

>> syms a b c d;

>> C1=[a b;c d]

C1 =

[ a, b]

[ c, d]

例3-19

>> x=solve('a*x^2 + b*x + c=0')

x =

-1/2*(b-(b^2-4*a*c)^(1/2))/a

-1/2*(b+(b^2-4*a*c)^(1/2))/a

>> syms a b c x

>> x=solve(a*x^2 + b*x + c)

x =

-1/2*(b-(b^2-4*a*c)^(1/2))/a

-1/2*(b+(b^2-4*a*c)^(1/2))/a

例3-20

>> [x,y]=solve('2*x^2+3*x+y','y^2-25*x')

x =

1

-2+3/2*i

-2-3/2*i

y =

-5

5/2+15/2*i

5/2-15/2*i

例3-21

>> y=dsolve('D2y-3*Dy+2*y=x','x')

y =

3/4+1/2*x+exp(2*x)*C1+exp(x)*C2

>> y1=dsolve('D2y-3*Dy+2*y=x','y(0)=1,y(1)=2','x')

y1 =

3/4+1/2*x-1/4*exp(2*x)*(exp(1)-3)/(exp(2)-exp(1))+1/4*exp(x)*(-3+exp(2))/(exp(2)-exp(1))

>> pretty(simple(y1))

exp(2 x) (exp(1) - 3) exp(x) (-3 + exp(2))

3/4 + 1/2 x - 1/4 --------------------- + 1/4 --------------------

exp(2) - exp(1) exp(2) - exp(1)

例3-22

>> int('x*sin(2*x)')

ans =

1/4*sin(2*x)-1/2*x*cos(2*x)

>> diff('x*sin(2*x)')

ans =

sin(2*x)+2*x*cos(2*x)

例3-23

>> syms a t s x

>> laplace(exp(-a*t))

ans =

1/(s+a)

>> L1=laplace(sin(x*t+2*t))

L1 =

(x+2)/(s^2+x^2+4*x+4)

>> f1=ilaplace(1/(s-a)^2)

f1 =

t*exp(a*t)

例3-24

>> syms n a w k z T

>> Z1=ztrans(cos(w*a*T))

Z1 =

(z-cos(a*T))*z/(z^2-2*z*cos(a*T)+1)

>> iztrans(Z1)

ans =

cos(a*T*n)

>> Z2=ztrans((k*T)^2*exp(-a*k*T))

Z2 =

T^2*z*exp(-a*T)*(z+exp(-a*T))/(z-exp(-a*T))^3

>> iztrans(Z2,k)

ans =

T^2*(1/exp(a*T))^k*k^2

第三章 程序

例3-1............................................................................................................................... 1

例3-2............................................................................................................................... 2

例3-3............................................................................................................................... 3

例3-4............................................................................................................................... 3

例3-5............................................................................................................................... 4

例3-6............................................................................................................................... 4

例3-7............................................................................................................................... 6

例3-9............................................................................................................................... 7

例3-10 . ............................................................................................................................ 7

例3-11 . ............................................................................................................................ 8

例3-12 . ............................................................................................................................ 8

例3-13 . ............................................................................................................................ 9

例3-14 . ............................................................................................................................ 9

例3-15 . .......................................................................................................................... 10

例3-16 . ...........................................................................................................................11

例3-17 . .......................................................................................................................... 12

例3-18 . .......................................................................................................................... 12

例3-19 . .......................................................................................................................... 13

例3-20 . .......................................................................................................................... 14

例3-21 . .......................................................................................................................... 14

例3-22 . .......................................................................................................................... 15

例3-23 . .......................................................................................................................... 15

例3-24 . .......................................................................................................................... 15

例3-1

>> A=[1 3 4]

A =

1 3 4

>> B=[1;3;4]

B =

1

3

4

>> C=[1 2 3;4 5 6]

C =

1 2 3

4 5 6

>> D=4.5

D =

4.5000

>> E=[]

E =

[]

>> C=[1 2 3;4 5]

??? Error using ==> vertcat

CA T arguments dimensions are not consistent.

例3-2

>> a=1:5

a =

1 2 3 4 5

>> t=0:pi/4:pi

t =

Columns 1 through 4

0 0.7854 1.5708 2.3562

Column 5

3.1416

>> b=pi:-pi/4:0

b =

3.1416 2.3562 1.5708 0.7854

>> CC=[C' C']

CC =

1 4 1 4

2 5 2 5

3 6 3 6

>> C(3,4)=10

C =

1 2 3 0

4 5 6 0

0 0 0 10

例3-3

>> yourName=input('请输入您的姓名：\n','s')

请输入您的姓名：

zhao

yourName =

zhao

>> yourAge=input('请输入您的年龄：')

请输入您的年龄：33

yourAge =

33

例3-4

>> A=magic(3)

A =

8 1 6

3 5 7

4 9 2

>> B=eye(size(A))

B =

1 0 0

0 1 0

0 0 1

>> t=logspace(0,5,6)

t =

Columns 1 through 4

1 10

Columns 5 through 6

10000 100000

例3-5

>> load -ascii mydata.dat

>> mydata

mydata =

16.0000 2.0000 3.0000

5.0000 11.0000 10.0000

9.0000 7.0000 6.0000

4.0000 14.0000 15.0000

-5.7000 -5.7000 -5.7000

-5.7000 -5.7000 -5.7000

8.0000 6.0000 4.0000

例3-6

>> A=magic(3)

A =

8 1 6

3 5 7

4 9 2

>> a22=A(2,2)

a22 = 100 13.0000 8.0000 12.0000 1.0000 -5.7000 -5.7000 2.0000 1000

5

>> A1=A(1:2:3,[2,3])

A1 =

1 6

9 2

>> A2=A(:,2:end)

A2 =

1 6

5 7

9 2

>> L1=logical([1 0 1]);

>> L2=logical([1 1 0]);

>> A3=A(L1,L2)

A3 =

8 1

4 9

>> A4=A;

>> A4(1,2)=0

A4 =

8 0 6

3 5 7

4 9 2

>> A4(4,4)=121

A4 =

8 0 6

3 5 7

4 9 2

0 0 0 0 0 0 121

>> A5=A;

>> A5(:,3)=[]

A5 =

8 1

3 5

4 9

>> A6=rot90(A)

A6 =

6 7 2

1 5 9

8 3 4

例3-7

>> clear

>> a = [2 4 6 5]; b = ones(4, 4) * 3; c =magic(4);

>> save -ascii mydat.dat

>> clear

>> load -ascii mydat.dat

>> mydat

mydat =

2 4 6 5

3 3 3 3

3 3 3 3

3 3 3 3

3 3 3 3

16 2 3 13

5 11 10 8

9 7 6 12

4 14 15 1

>> clear

>> a = [2 4 6 5]; b = ones(4, 4) * 3; c =magic(4);

>> save mydat

>> clear

>> load mydat

>> whos

Name Size Bytes Class Attributes

a 1x4 32 double b 4x4 128 double c 4x4 128 double

例3-9

>> A=magic(3)

A =

8 1

3 5

4 9

>> B=eye(3)

B =

1 0

0 1

0 0

>> C=A+B

C =

9 1

3 6

4 9

例3-10

>> D=A*C

D =

99 68

70 96

71 76

>> D1=A.*C

D1 =

6 7 2 0 0 1 6 7 3 73 74 93

72 1 36

9 30 49

16 81 6

>> D2=3*magic(3)

D2 =

24 3 18

9 15 21

12 27 6

例3-11

>> A =[8 1 6; 3 5 7; 4 9 2];

>> B=[1 2 3]';

>> X=A\B

X =

0.0500

0.3000

0.0500

例3-12

>> A=magic(3);

>> B=2;

>> C=A^B

C =

91 67 67

67 91 67

67 67 91

>> B^A

ans =

1.0e+004 *

1.0942 1.0906 1.0921

1.0912 1.0933 1.0924

1.0915 1.0930 1.0923

例3-13

>> A = [1 1 1;1 2 3;1 3 6]

A =

1 1 1

1 2 3

1 3 6

>> X = A.^2

X =

1 1

1 4

1 9

>> X1=2.^A

X1 =

2 2

2 4

2 8

>> C=X.^B

C =

1

1

1

例3-14

>> A=magic(3)

A =

8 1

3 5

4 9

>> A'

1 9 36 2 8 64 1 16 81 6 7 2 1 81 1296

ans =

8 3 4

1 5 9

6 7 2

>> B=i*eye(3)

B =

0 + 1.0000i 0 0 0 0 + 1.0000i 0 0 0 0 + 1.0000i

>> C=(A+B)'

C =

8.0000 - 1.0000i 3.0000 4.0000 1.0000 5.0000 - 1.0000i 9.0000 6.0000 7.0000 2.0000 - 1.0000i

>> C=(A+B).'

C =

8.0000 + 1.0000i 3.0000 4.0000 1.0000 5.0000 + 1.0000i 9.0000 6.0000 7.0000 2.0000 + 1.0000i

例3-15

>> A=magic(3)

A =

8 1 6

3 5 7

4 9 2

>> B=[3 4 5;6 9 7;1 8 2]

B =

3 4 5

6 9 7

1 8 2

>> A>=B

ans =

1 0 1

0 0 1

1 1 1

>> A~=B

ans =

1 1 1

1 1 0

1 1 0

例3-16

>> A=[1 0 2;3 0 5]

A =

1 0 2

3 0 5

>> B=[2 6 1; 0 0 8]

B =

2 6 1

0 0 8

>> A&B

ans =

1 0 1

0 0 1

>> A|B

ans =

1 1 1

1 0 1

>> ~A

ans =

0 1 0

0 1 0

>> xor(A,B)

ans =

0 1 0

1 0 0

例3-17

>> str=input('input a string:\n','s');

if ~isempty(str) && ischar(str)

sprintf('Input string is ''%s''', str)

end

input a string:

MA TLAB is great

ans =

Input string is 'MATLAB is great'

例3-18

>> clear

>> f=sym('a*x^2+b*x+c')

f =

a*x^2+b*x+c

>> f-a

??? Undefined function or variable 'a'.

>> syms a b c x

>> f1=a*x^2+b*x+c

f1 =

a*x^2+b*x+c

>> f1-a

ans =

a*x^2+b*x+c-a

>> clear

>> A=sym('[a b;c d]')

A =

[ a, b]

[ c, d]

>> C=[a b;c d]

??? Undefined function or variable 'a'.

>> syms a b c d;

>> C1=[a b;c d]

C1 =

[ a, b]

[ c, d]

例3-19

>> x=solve('a*x^2 + b*x + c=0')

x =

-1/2*(b-(b^2-4*a*c)^(1/2))/a

-1/2*(b+(b^2-4*a*c)^(1/2))/a

>> syms a b c x

>> x=solve(a*x^2 + b*x + c)

x =

-1/2*(b-(b^2-4*a*c)^(1/2))/a

-1/2*(b+(b^2-4*a*c)^(1/2))/a

例3-20

>> [x,y]=solve('2*x^2+3*x+y','y^2-25*x')

x =

1

-2+3/2*i

-2-3/2*i

y =

-5

5/2+15/2*i

5/2-15/2*i

例3-21

>> y=dsolve('D2y-3*Dy+2*y=x','x')

y =

3/4+1/2*x+exp(2*x)*C1+exp(x)*C2

>> y1=dsolve('D2y-3*Dy+2*y=x','y(0)=1,y(1)=2','x')

y1 =

3/4+1/2*x-1/4*exp(2*x)*(exp(1)-3)/(exp(2)-exp(1))+1/4*exp(x)*(-3+exp(2))/(exp(2)-exp(1))

>> pretty(simple(y1))

exp(2 x) (exp(1) - 3) exp(x) (-3 + exp(2))

3/4 + 1/2 x - 1/4 --------------------- + 1/4 --------------------

exp(2) - exp(1) exp(2) - exp(1)

例3-22

>> int('x*sin(2*x)')

ans =

1/4*sin(2*x)-1/2*x*cos(2*x)

>> diff('x*sin(2*x)')

ans =

sin(2*x)+2*x*cos(2*x)

例3-23

>> syms a t s x

>> laplace(exp(-a*t))

ans =

1/(s+a)

>> L1=laplace(sin(x*t+2*t))

L1 =

(x+2)/(s^2+x^2+4*x+4)

>> f1=ilaplace(1/(s-a)^2)

f1 =

t*exp(a*t)

例3-24

>> syms n a w k z T

>> Z1=ztrans(cos(w*a*T))

Z1 =

(z-cos(a*T))*z/(z^2-2*z*cos(a*T)+1)

>> iztrans(Z1)

ans =

cos(a*T*n)

>> Z2=ztrans((k*T)^2*exp(-a*k*T))

Z2 =

T^2*z*exp(-a*T)*(z+exp(-a*T))/(z-exp(-a*T))^3

>> iztrans(Z2,k)

ans =

T^2*(1/exp(a*T))^k*k^2


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