Manual Maxima - Vallejo

7/26/2019 Manual Maxima - Vallejo

1/98


2/98
http://galia.fc.uaslp.mx/~jvallejo


3/98

n > 2

(A)< 0

(A)2 4 (A)> 0 (A)2 = 4 (A)> 0

(A)2 0

a= 0

a = 0


4/98


5/98


6/98

sin(x) cos(x) x

sin(x) sen(x)

y = y(x)


7/98

x

y(x) = dydx

.

y y y y (n)

y(n)(x) = dny

dxn.

t

y(t) =dy

dt, y(t) =

d2y

dt2
http://www.math.utexas.edu/pipermail/maxima/


8/98

( %o1) 64

5

%

%

% % % %

( %o2) 1

( %o3)

4

( %o4) 2

2

% % % %


9/98

( %o5) 1.414213562373095

( %o6) 25

22

=

2

( %o7) 1.414213562373095048801689b0

bN 10N b3 103

( %o8) 1.001414213562373095048802b3

3 + 2i 1 i i

( %o9) 2 i + 3

( %o10) i 1


10/98

( %o11) i + 2

( %o12) 1

( %o13) 2

z0

( %o14) (i 1) (2 i + 3)

( %o15) e(i1)(2 i+3)

( %o16) (i 1) (2 i + 3)

( %o17) 5 i 1


11/98

( %o18) 5 i 1

( %o19) e5 i1

( %o20) e1 (cos (5) isin (5)) ,

( %o21) 2 e3 isin (2) + 2 e3 cos (2)

( %o22)

2 e3 i 4

( %o23) z0= (

i

1) (2 i + 3)

( %o24) z1 = 5 i 1


12/98

( %o25) 4

b+

2

a

( %o26) 2

b + 4 a

a

b

( %o27) a

b+

a

b

( %o28)

ab

b + a

b

( %o29)

a

b + a

b

( %o30)

abc


13/98

( %o31) ab c

12!

( %o32) 210 35 52 7 11

( %o33) b4 + 4 a b3 + 6 a2 b2 + 4 a3 b + a4

( %o34) [x=

b2

4 a c + b

2 a , x=

b2

4 a c

b

2 a ]

( %o35) [[x= 1, y= 0]]

a

( %o36) [[x= 13

5 a + 3, y = a 2

5 a + 3]]


14/98

( %o37) [x= 2.0 i+.9999999999999999, x= .99999999999999992.0 i, x= 3.0]

( %o38) [x= 3]

( %o39) [x=

6, f(x) = 1]

0

( %o40) [x=

6, f(x) = 1]

(x) = x2 (x) x2 = 0 x x0 = 0.5 0.001


15/98

( %o42) .8241461317281952

( %o43)

( %o44)


16/98

x y

( %o46) [[x= 3.756834008012769, y= 2.779849592817897]]

lma2 cos a

( %o47) 2

+

( %o48) 0

( %o49) und


17/98

( %o50) 2 xsin x2

cos(x2)

( %o51) 2 sin x2 4 x2 cos x2

( %o52) d2

d x2cos

x2

= 2 sin x2 4 x2 cos x2

( %o53)atan

3x6

6

( %o54) 3 e

5 3

5

( %o55)

0

ex cos (x)2dx=3 e

5 3

5


18/98

xy(1 + y2) x (1 + x2) y= 0

xy(1 + y2) (1 + x2) y x

= 0.

( %o1) x y

y2 + 1 x2 + 1 d

d xy

= 0

( %o2) log y2 + 1 2

log (y)

2 =

log x2 + 12 + %c

y

x

y = y(x) 0

%c

x = x0, y = y0 x0 y0

y= y(x) y(1) = 1


19/98

( %o3) log(y2

+ 1) 2 log(y)2

= log(x2

+ 1) 2 log(2)2

log(A) + log(B) log(AB)

log(ab) a log(b) log(ab) log(a)+log(b)

12log(a) log(

a)

( %o6) log( |y|

y2 + 1) = log(

x2 + 1

2 )

c c = log(k)

(1 + x2)(1 + y2) = ky2,

y2

y2 + 1=

x2 + 1

4 .

y

( %o7) [y =

x2 + 1

3 x2

, y=

x2 + 1

3 x2

]


20/98

( %o8) separable

xy(1 + y2) (1 + x2) y

x= 0

ex sec(y)dx + (1 + ex) sec(y) tan(y)dy= 0,

y0 = 3

x0 = 3

( %o9) (ex + 1) sec (y)tan (y)

d

d xy

+ ex sec (y) = 0

( %o10) log (sec (y)) = log (ex + 1) + %c

( %o11) log (sec (y)) = log (ex + 1) log

e3 + 1

log (2)

( %o12) log (sec (y)) = log

ex + 1

2 e3 + 2

( %o13) [y = asec

2 e3

ex + 1+

2

ex + 1

]

( %o14) separable


21/98

2xydx (x2 y2)dy= 0

( %o15) 2 x y x2 y2 dd x

y

= 0

( %o16

) y

y2 + x2 = %c

( %o17) [y=

1 4 %c2 x2 + 12 %c

, y=

1 4 %c2 x2 1

2 %c ]

( %o18

) exact

M(x, y)dx + N(x, y)dy= 0

M

y =

N

x :

( %o19) 0

( %o20) (b x a y)

d

d xy

b y+ a x= 0


22/98

( %o21) %c x= e(b+a) log( y+xx )+(ab) log(

yxx )

2 a

( %o22) homogeneous

( %o23) %c x= x (y x)ba

2 a

(y+ x)b+a2 a

x

k= c2a k= (y+ x)a+b(y x)ab.

( %o24)

y2 + 6 x y+ 20 x2 d

d xy

+ 4 y2 + 2 x y+ 3 x2 = 0

( %o25) %c x= x (y+ 3 x)

y2 + 7 x y+ x2

( %o26) homogeneous

[y=

45 %c2 x2 2 %c x + 1 + 7 %c x 12 %c

,

y=

45 %c

2 x2 2 %c x + 1 7 %c x + 12 %c

]


23/98

M(x, y)dx +N(x, y)dy = 0

(x, y)

Mdx + Ndy = 0

(M)

y =

(N)

x .

xdy+ ydx = x2y2dx

( %o28) x

d

d xy

x2 y2 y= 0

( %o29) x3 y+ 3 x

3 y = %c

( %o30) exact

( %o31) 1

y2

( %o32) [y= 3 x

x3 + 3 %c ]

y2

2 + 2yex

dx + (y+ ex)dy= 0 :


24/98

( %o33) d

d xy =

y22 2 ex yy+ ex

( %o34) ex y2 + 2 e2 x y= %c

( %o35

)[y= ex

ex2

e3 x + %c + e2x

, y = ex

ex2

e3x + %c e2x

]

( %o36) exact

( %o37) ex

dy

dx+ yP(x) = Q(x).

Q(y) = 0

y(x) = e

P(x)dx

c +

Q(x)eP(x)dxdx

.

y+ 2xy= x


25/98

( %o38) d

d xy + 2 x y= x

( %o39) y= ex2

ex

2

2 + %c

( %o40) linear

y+ sin(x)y= 2xecos(x)

( %o41) d

d xy + sin (x) y = 2 ecos(x)

( %o42) y= (2 x + %c)ecos(x)

( %o43) linear

y

x2y + xy =x4 + x2

( %o44) x2

d

d xy

+ x y= x4 + x2


26/98

( %o45) y=(x

2

+1)2

4 + %c

x

( %o46) linear

y+ P(x)y= Q(x)yn,

y 3x

y= y5

( %o47) d

d xy 3 y

x =y5

( %o48) y= x3

%c 4x

13

13 14

( %o49) bernoulli

x

dy

dx

2 (1 + xy) dy

dx+ y= 0.


27/98

( %o51) x

d

d xy

2 (x y+ 1)

d

d xy

+ y= 0

x

d

d xy

2 (x y+ 1)

d

d xy

+ y= 0

[y = log (x) + %c, y = %c ex]

( %o53) factor

dydx

2 + x dy

dx y = 0

( %o54)

d

d xy

2+ x

d

d xy

y= 0

dd x

y2

+ x dd x

y y= 0

[y= %c x + %c2, y= x

2

4]


28/98

( %o56) clairault

dy

dx= (x + y)2 :

( %o57) d

d xy = (y+ x)

2

( %o58)

[[x= %catan

%t

, y= x

%t], [x= atan

%t

+ %c, y=

%tx]]

( %o59) lagrange

dy

dx (1 2x)y+ y2 = 2x.

erf(x) =

x

0 et2

dt

( %o60) d

d xy + y2 (1 2 x) y = 2 x

[y =ex

ex

%c ex2+ 14 erf

2 x+1

2 +

ex

2

+ 2 %c

%c ex2+ 14 erf2 x+12 + ex2 ]

( %o62) riccati


29/98

d2x

dt2 + a1(t)

dx

dt + a2(t)x= b(t),

a1, a2, b b(t) 0

%k1, %k2 x+ x 2x= 0

( %o63) d2

d t2x +

d

d tx 2 x= 0

( %o64) x= %k1 et + %k2 e2 t

( %o65) constcoeff

x x= t2et

( %o66) d2

d t2x d

d tx = t2 et

( %o67) x=

t3 3 t2 + 6 t 6 et

3 + %k1 et + %k2


30/98

( %o68) variationofparameters

y(x0) = y0 y(x0) = y0

y 6y+ 9y = 0 x0 = 0 y0 = 1 y0 =1

( %o69) d2

d x2y 6

d

d xy

+ 9 y= 0

( %o70) y= ( %k2 x + %k1)e3x

( %o71) y= (1 4 x)e3x

k1 k2

x = 0 y = 2 x= 3 y= 4

( %o72

) y=

2 e92 e9

4 x3

e

3x


31/98

( %o73) d2

d t2x + (1 t)

d

d tx

t x= 0

( %o74) false

{et, et 12 erf

i t2

+ i

2

}

et

4y 4y(x) y x

t

x = x 4y 6t + 3y = x + 3y+ 3tz= 8x + 4y 2z


32/98

( %o76) d

d tx (t) = 4 y (t) x (t) 6 t + 3

( %o77) d

d ty (t) = 3 y (t) + x (t) + 3 t

( %o78) d

d tz (t) = 2 z (t) + 4 y (t) 8 x (t)

[x(t) = 4y(0)tet 2x(0)tet 6tet + (x(0) + 3)et + 6t 3,y(t) = 2y(0)tet + x(0)tet + 3tet + y(0)et 3t,

z(t) =40y(0)tet

3

+20x(0)tet

3

+

20tet (28y(0) + 44x(0) + 132)et

9 +

(9z(0) + 28y(0) + 44x(0) 111)e2t9

30t + 27]

x(0) y(0) z(0)

x(0) = 0 y(0) = 1 z(0) =1

i = 1,...,n

x(0) = 0 y(0) = 1 z(0) =1


33/98

[x(t) = 10tet

+ 3et

+ 6t 3,y(t) = 5tet + et 3t,

z(t) =100tet

3 160e

t

9 74e

2t

9 30t + 27]

b expr x a

x = 3x + 8z+ ty = 3x y+ 6z+ tz= 2x 5z+ t

x(0) =y(0) =z(0) = 1,

( %o81)

d

d tx (t) = 8 z (t) + 3 x (t) + t

( %o82) d

d ty (t) = 6 z (t) y (t) + 3 x (t) + t

( %o83) d

d tz (t) = 5 z (t) 2 x (t) + t

( %o84) 1


34/98

( %o85) 1

( %o86) 1

[x (t) = 24 t et + 26 et + 13 t 25,y (t) = 18 t et + 20 et + 10 t 19,z (t) = 12 t et 10 et 5 t + 11]

x(0) y(0) z(0)

x(t) y(t) z(t) x(t)

( %o88) done

( %o89) x (0)

x

x(t) x(t) x

n > 2

n >2

an(t)dnx

dtn + + a1(t) dx

dt + a0(t)x= b(t),


35/98

2

d3y

dx3+

d2y

dx2+

dy

dx+ y x= 0

( %o94)

y(x) =sin (x)

d2

d x2y (x)

x=0

+ 2

dd x

y (x)

x=0

+ y (0) 1

2

+

ex

d2

d x2y (x)

x=0

+ y (0) + 1

2

cos (x) d2d x

2 y (x)x=0

y (0)

12

+ x 1

y+ y 2y = 5 ex

y(0) = 0 = y(0) y(0) = 1

( %o96) d3

d x3y (x) +

d2

d x2y (x) 2

d

d xy (x)

= 5 ex


36/98

( %o97) 1

( %o98) 0

( %o99) 0

( %o100) y (x) = x ex

3 +

22 ex

9 7 e

2x

36 5 x

2 9

4

( %o101) y (x) = e2 x(12 x 88)e3 x + (90 x + 81)e2x + 736

y y

( %o102) done


37/98

x(t) = f(x, t)

x= (t) x= (t) x(t) Rn f : Rn R Rn x= (t)

>0 x= (t) ||(t0) (t0)|| <

||(t) (t)|| < , t > t0

x= (t) >0


38/98

x= (t) ||(t0) (t0)|| <

||(t) (t)|| 0, si t 0.

t 0

n

x(n)(t) = f(t, x(t), x(t),...,x(n1)(t),

x1(t) := x(t), x2(t) := x(t),...,xn(t) := x(n1)(t),

x1(t) = x2(t),x2(t) = x3(t),

xn1(t) = xn(t),xn(t) = a0(t)x1(t) a1(t)x2(t) an1(t)xn(t) + b(t).


39/98

x= x + y

y= y.

x(0) = 0 y(0) = 0 (0, 0)

( %o1

)

d

d tx (t) = y (t) + x (t)

( %o2) d

d ty (t) = y (t)

( %o3) [x (t) = y (0)t et + x (0)et, y (t) = y (0)et]

( %o4) [x (t) = 0, y (t) = 0]

103 t= 0 x y (x(0), y(0)) =(0.001, 0.001) 103

(x(0), y(0)) = (0.001, 0.001) 0.001

(x(0), y(0)) = (0, 0) (x(0), y(0)) = (0.002, 0.002) (0, 0) %

(x(0), y(0)) = (0.002, 0.002)


40/98

( %o5) [x (t) = 0.002 t et + 0.002 et, y (t) = 0.002 et]

x, y t t= 5

( %o6) [x (5.0) = 1.780957909230919, y (5.0) = .2968263182051532]

x = 0 x(t) = k k R = (t) k1 (t) k2 ||(t)(t)|| = |k1k2|

t x(t) =f(x) x(t) = x x

f(x) = 0 x = 0 x = x2

(t) t0 x= 0 t t0 t < t0 t > t0

( %o7)

1

x

=t + %c

( %o8) [x= 1t + %c

]


41/98

c

{1,

1/2, 0, 1/2, 1

}

[1.5, 1.5]

( %o9)

x = 0

t

t= i2

x= 1 x= 1x2 x= 1


42/98

( %o11) log (x + 1)

log (x

1)

2 =t + %c

x = 1 x= 1 t= 0 log(0)

x = 1

t = 1 x = 1 t= 1 1.01 1.001 1.0001

[ log (x + 1) log (x 1)2

=2 t + log (100) + log201100 2

2 ,

log (x + 1) log (x 1)2

=2 t + log (1000) + log

20011000

22

,

log (x + 1) log (x 1)2

=2 t + log (10000) + log

2000110000

22

]

[

log

x1x+1

2

=2 t + log (201) 2

2 ,

log

x1x+1

2

=2 t + log (2001) 2

2 ,

log

x1x+1

2

=2 t + log (20001) 2

2 ]

x 1x + 1

= exp(2t log(2 10i + 1) + 2),

( %o14) [x= e22 t + 210i + 1

e22 t 2 10i 1 ]

x= 1


43/98

( %o15)

= 0.01

< 0.01

x= 1

x= 1

x(t) = A(t)x(t) + b(t),

x(t) Rn

t [t0, t1] A(t) Matnn(R)

n


44/98

x= (t) x

0

x(t) = A(t)x(t)

A (A(t)) = 0 t x= 0

A(t) = A A

x(t) = A(t)x(t) + b(t)

x= y+ cos(t)y= x + sin(t).

( %o16)[x (t) = (y (0) + 1)sin (t) + x (0)cos (t) , y (t) = y (0)cos (t) x (0)sin (t)]

| sin(t)|, | cos(t)| 1 (x(0), y(0))

|x(t)| |y(0) + 1| + |x(0)||y(t)| |y(0)| + |x(0)|


45/98

A

t

A

x(t) = Ax(t) + b(t).

A

A

A

A

mA(x) = xn + cn1xn1 + + c1x + c0 A mA(A) = 0

A

A(x) = (x1)n1

(xk)nk

mA(x) = (x1)m1 (xk)mk 1 mi ni

A(x) = mA(x) a R

x(t) = Ax(t) + b(t)

A=

3 a 2 13 (2 + a) 0

a 3 a 3 4

( %o17)

3 a 2 13 a 2 0

a 3 a 3 4


46/98

A(x)

( %o19)(x 4) (x a 2) (x a + 3)(a 3) (x a 2)+6 (x 4)3 (a 3)

( %o20) (x + 3) (x + a 1) (x + a + 1) 1 =

3 2 = 1

a

3 =(1 +a)

A

( %o21) [[3, a 1, 1 a], [1, 1, 1]],

A(x) = mA(x)

( %o23) (x + 3) (x + a 1) (x + a + 1)

1 =3 2 = 1 a 3 = (1 + a)


47/98

( %o25) [1< a]

1< a

1 a= 0 (1 + a) = 0 a = 1 a =1 2 = 2 > 0 2 = 0 3 =2

a > 1

a= 1

a < 1


48/98

K

A=

a11 a12 a1na21 a22 a2n

an1 an2 ann

APAP P A 2 2

a bc d

( %o1)

a bc d

( %o3) (A) = a d + (d a) b c + 2

(A) = a + d (A) = ad bc

(A) = 2 (A) + (A).

A (A) (A)

1= 2 A P

1 00 2


49/98

(A)2 4 (A)> 0.

( (A), (A)) (A) = 14 (A)

2

(A)2 4 (A) > 0 (A) =14 (A)

2 1, 2

(A)2 4 (A) = 0 (A) = 14 (A)2 1 = 2 R

(A)

2

4

(A) < 0

(A) =14 (A)

2 1, 2

( i)

( %o4) 2 2 1 + 1 2 %

(A) = 1+ 2,

(A) = 12.

x(t) = Ax(t)

x = (x, y)

A

x = x(t) y = y(t) y= y(x) t y = y(x) (x, y)

x(t) = F(x,x)

(x0, x0) (x,x)

x= y y= F(x, y)


50/98


51/98

( %o10) [y=y0 x

21

x210

]

y = y0

x

x0

21

x= 3x 2yy= 4x 3y

( %o11) [[1, 1], [1, 1]]

1 =1 2 = 1

y= y0x0

x

P

(x0, y0)


52/98

( %o13)

( %o13)

(A)2 4 (A) > 0

1 2 > 0


53/98

1, 2 < 0 1, 2 > 0

21

1, 2 < 0 1, 2 > 0

x= y

y = 2 x + 3 y

( %o15) [[1, 2], [1, 1]]

1 = 1 2 = 2

y= y0

x

x0

2

( %o16)


54/98

( %o17)

x= yy= 2 x 3 y

( %o19) [[2, 1], [1, 1]]


55/98

( %o20)

( %o21)


56/98

(A)2 = 4 (A) > 0

1 = 2 = R < 0 > 0

A

P

00

A

= 1

( %o22)

>0

< 0

x= xy= y


57/98

A

P

10

( %o23

)

d

d tx (t) = y (t) + x (t)

( %o24) d

d ty (t) = y (t)

( %o25) [x (t) = y (0)t e t + x (0)e t, y (t) = y (0)e t]

( %o26) [t= e tx0 e t x

y0]

( %o27

) [t=

log y

y0 ]


58/98

( %o28) [y= y0 e e t x

y0 x0

y0 ]

x= y

y= x + 2y

( %o30) [[1, 2]]

1 10 1

( %o32)


59/98

( %o33)

< 0

(A)2 < 4 (A), (A) > 0

1 =a + bi 2 = a bi b = 0 a < 0 a >0 a= 0

A

P A

a bb a

( %o34) d

d tx (t) = a x (t) b y (t)


60/98

( %o35) d

d ty (t) = a y (t) + b x (t)

( %o36) [notequal (b, 0)]

[x (t) = x (0)ea t cos (b t) y (0)ea t sin (b t) ,y (t) = x (0)ea t sin (b t) + y (0)ea t cos (b t)]

a= 0

bi

( %o38)

[x (t) = x (0)cos (b t) y (0)sin (b t) , y (t) = x (0)sin (b t) + y (0)cos (b t)]

x= x + 2y

y= x y

( %o39) [[i, i], [1, 1]]

b= 1


61/98

( %o9

) [x (t) = x (0)cos (t) y (0)sin (t) , y (t) = x (0)sin (t) + y (0)cos (t)]

( %o40)

( %o41)


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a = 0

a bi

a > 0 x= 2x + 2y

y= x

( %o42) [[1

i, i + 1], [1, 1]]

a bi= 1 i 1 11 1

( %o43)


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( %o44)

x= 2x 2yy= x

( %o45) [[i 1, i 1], [1, 1]]

( %o46)


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( %o47)


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x= f(x, y)

y = g(x, y)

(x, y) f(x, y) = 0 = g(x, y) (x(t), y(t)) = (x, y)

V(x, y) (x, y)

z = V(x, y) (x, y)

V(x, y) z(t) = V(x(t), y(t))

dz

dt 0.

(x(t), y(t)) t= t0 (x0, y0) V(x, y) (x(t), y(t)) (x, y) (x, y)


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n

V

V(x, y)

V

x y V E

x = (0, 0)

x= 2xyy= x2 y3

E(x, y) = x2p + y2q.

E(x(t), y(t)) x =2xy y = x2 y3 t

( %o1) [x (t) , y (t)]

( %o2) x (t)

( %o3) y (t)

( %o4) E(x, y) := x (t)2p

+ y (t)2 q

dEdt


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, = 0 x= x + 2yy= x y

( %o11) 1 2

1

1

( %o13)

x

2 i + 1

x +

2 i + 1

1 =1 +

2i 2 =1

2i

, = 0

, = 0 (0, 0)

( %o14) [x (t) , y (t)]

( %o15) x (t)

( %o16) y (t)


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( %o17) E(x, y) := a x (t)2p + b y (t)2 q

2 b q y (t)2 q1

(y (t) + x (t))

y2 + x2 1+2 a p x (t)

2p1(2 y (t) x (t)) y2 x2 + 1

p = 1 = q xy2q1 x2p1y

2 b y (t) (y (t) + x (t))

y2 + x2 1+2 a x (t) (2 y (t) x (t)) y2 x2 + 1

xy

2byx(y2 + x 1) 4axy(y2 + x 1) = 0.

b= 2a

( %o18) 2 a

2 y (t)2

+ x (t)2

y2 + x2 1

y2 + x2 1 0 , 0

a 0 b= 2a

b= 2a= 2 = A = B A, B 0

( %o19) V (x, y) = 2 y (t)2

+ x (t)2


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( %o23) dd zV = 2

2 y (t)2 + x (t)2

y2 B+ x2 A + 1

V(x, y)0 dVdt

0 , 0 >0 >0

Fe=k sin k >0)

Fa =

a a >0 k = g/l g = 10ms2 l = 1m k = 10s2

k = 1

+ a+ sin = 0,

= x = y x = yy = ay sin x .

a

(xe, ye)

x = f(x, y)y = g(x, y)

f(xe, ye) = 0 = g(xe, ye) f g (xe, ye)

x = D1f(xe, ye)(x xe) + D2f(xe, ye)(y ye)y = D1g(xe, ye)(x xe) + D2g(xe, ye)(y ye) .

x xe x, y ye y

xy

=J(xe,ye)F

xy


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F(x, y) = (f(x, y), g(x, y)) J(xe,ye)F (xe, ye) (xe, ye) J(xe,ye)F

(xe, ye) (xe, ye)

(0, 0)

(xe, ye) (0, 0)

f(x, y) = y g(x, y) = ay+ sin x

y= 0ay sin x= 0 ,

(xe, ye) = (0, 0)

F(x, y) = (y, ay sin(x))

0 1

cos (x) a

0 11 a


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0 11 a

( %o13) [[

a2 + 4 + a

2 ,

a2 + 4 a

2 ], [1, 1]]

(, 0)

( %o14)

a


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( %o15)

( %o15)


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x= f(t, x), t > t0

x(t0) = x0

x(t) x(t) x(t0) = x0 t0 = 0 x(t) t0, t1,...,tn x= x(t) [t0, T] n

T =t0+ nh

h h= Tt0n

xj x(tj ) = x(t0+jh).

j = 1,...,n x0 = x(t0)

[tj , tj+1]

vj =f(t, vj ), t > tjvj (tj ) = xj

vj (t) xj tj tj+1


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vj (tj+1)

xj j = 1,...,n

vj (tj+1) xj+1 xj

xk k j

xj x0

xj+1 xj xk k j xj+1 xj+1 vj (tj+1)

xj+1 = F(xj , xj+1,...,xj

l)

G(xj+1, xj ,...,xjl) = 0,

xj+1


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xj+1 xj k xj+1 k

tj+1

L(tj+1) = vj (tj+1) xj+1.

tj+1

G(tj+1) = x(tj+1)

xj+1.

j

L(tj+1) 0

h 0 T G(tj+1) 0 h0 p >0 O(L) =O(h

p) p

p vj tj f(x, t)

consistencia + estabilidad convergencia.

p 1

dx

dt x= f(x, t). x(t0),...,x(tn) [ti, ti+1] f(x, t) t x(tn)


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f(x, t)

h

x(tn) = lmh0

x(tn+ h) x(tn)h

= lmh0

xn+1 xnh

h

xn+1 xn+ hx(tn),

xn+1 xn+ hf(tn, xn).

xn+1 = xn+ hf(tn, xn)

x(t0) =x0 x(tn) h tn x(tn) x(t)

x= a x a R

h 0

xn+1 = xn+ h(a xn) = (1 h)xn+ ha.

xn x0 h a

( %o2) xn= a (1 h)n + %k1(1 h)n + a


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k1 t= 0 k1 = x0

xn= x0(1 h)n a(1 h)n + a

[0, t] t > 0 h = tn

xn= x0(1 tn

)n a(1 tn

)n + a

h0 n

( %o3) [t > 0]

( %o4) a et + x0 et + a

( %o5) et

a et a + x0

( %o6) x= et

a et + %c

( %o7) x= et

a et a + x0


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ti xi =x(ti)

h

h 0 h= 0.3, 0.2, 0.1 a = 0.5 x(0) = 1 h = 0.3 h = 0.2 h = 0.1 h

t [0, 6] 6 = hn h = 0.3 20 h= 0.1 n= 60

h= 0.3

( %o8) 0,3

( %o9) 1

( %o10) x1n:= x1n1 h1 x1n1+ 0,5 h1

$

h= 0.2

( %o12) 0,2


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( %o13) 1

( %o14) x2n:= x2n1 h2 x2n1+ 0,5 h2

h= 0.1

( %o16) 0,1

( %o17) 1

( %o18) x3n:= x3n1 h3 x3n1+ 0,5 h3

a = 0.5 60 h = 0.3


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( %t22)

( %o22) [gr2d (points, points, points, points)]

%

vj =f(t, vj ), t > tjvj (tj ) = xj,

vj (tj+1) vj (jj ) = tj+1

tj

f(s, vj (s)) s.


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[tj , tj+1] f(tj , vj (tj )) f(tj+1, vj (tj+1))

xj+1 xj = h2

(f(tj , xj ) + f(tj+1, xj+1))

3 2

hf(tj , xj )

2

xj+1

xj+1 = xj+ hf(tj , xj )

K1 = hf(tj , xj )


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K2 = hf(tj+1, xj+ K1)

2

xj+1 = xj+1

2(K1+ K2).

p = 3 2 f

4 4 f 4

K1 = hf(tj , xj )

K2 = hf(tj+ 12 , xj+1

2K1)

K3 = hf(tj+ 12 , xj+1

2K2)

K4 = hf(tj+1, xj+ K3)

tj+ 12 tj+ 12 = 12(tj+1 tj )

xj+1 = xj+1

6(K1+ 2K2+ 2K3+ K4).

p = 5 4

4

p = 5 f

m

p(m) = m, para m= 1, 2, 3, 4

p(m) = m

1, para m= 5, 6, 7

p(m) = m 2, para m= 8, 9p(m) m 2 para m 10


86/98

x= 2t

et2 x

.

(t= 0, x= 2)

4

( %o25) d

d tx = 2 t

et

2 x

( %o26) x=

t2 + %c

et2

( %o27) x=

t2 + 2

et2

.

, x , x0, [t, t0, T , h] x(t) = f(x, t) x t

x0 x(t) t= t0 T

h

[t0 = 0, T = 5]

h= 0.2 (tj , xj )


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( %o29) [gr2d (explicit, points)]

y = y + sin(x),

y = 1 x = 0

( %o30) d

d xy = y + sin (x)


88/98

( %o31) y= ex

ex (sin (x) cos (x))2

+ %c

( %o32) y= sin (x) + cos (x) 3 ex

2

h= 0.1



89/98

[0, 0.5] h h = 0.2 [0, 5]

4

x = 1

n

2

y y= 2 cos(x)ex

(x = 0, y = 1) (x = 0, y = 0) [0, 3] x1(t) = y(t) x2(t) = y(t)

x1 = x2

x2 = x1+ 2 cos(t)et.

( %o35) d2

d x2y y= 2 ex cos (x)

( %o36) y=4 ex sin (x) 2 ex cos (x)

5 + %k1 ex + %k2 ex

( %o37) y=4 ex sin (x) 2 ex cos (x)

5 +

ex

2 +

9 ex

10


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h = 1

(tj , (x1)j , (x2)j ) (tj , yj, yj )

(tj , yj )


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y y= ecos(x).

(x = 0, y = 1) (x = 0, y = 0) ecos(x)

x1 = x2

x2 = x1+ ecos(t).

( %o43)


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Manual Maxima - Vallejo