AP Ecuaciones Diferenciales Ordinarias
Transcript of AP Ecuaciones Diferenciales Ordinarias
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 1/21
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 2/21
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 3/21
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 4/21
hi = xx+1 − xi x
y = f (x, y)
y(n) = f (x,y,y
, y
,....,y(n+1))
y = f (x, y)
y = y(x)
y = f (x, y(x))
y = y(x)
y = f (x, y(x)) (xi, yi)
y = y(x)
(xi, yi)
(x, y)
x
dy
dx = x
dy =
x dx ⇒ y =
x2
2 + C
P = (x0, y0)
x0
P
y = x
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 5/21
y = f (x, y(x)) D xy
L > 0
|f (x, y2) − f (x, y1)| ≤ L|y2 − y1|
D
y = f (x, y)
f (x, y)
D
xy
D
(x0, y0) ∈ D y(x)
y = f (x, y(x))
y(x0) = y0
h → 0
h → 0
h → 0
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 6/21
n
n
t = 0
y = y (x)
y
= f (x, y(x))
y = y(x)
m + 1
x0
x0
y(x) = y(x0)+y(x0)(x−x0)+y(x0)
2! (x−x0)2+
y(x0)
3! (x−x0)3+ ...+
ym(x0)
m! (x−x0)m+
ym+1(ξ )
(m + 1)!(x−x0)m+1
ξ ∈ (x0, x)
y = f (x, y(x)) y = f x + f · f y y = f xx + 2f · f xy + f 2 · f xy + f x · f y + f · f 2y
f x f x f y f y
h
xn = x0 + n · h n = 0, 1, 2, 3...
m = 2
yn+1 = yn + hf (xn, yn) + h2
2! (f x + f · f y)2 + ... +
h3
3! f (2)(ξ,y(ξ))
ξ ∈ (xn, xn+1) h = xn+1 − xn
y = −y + x + 1 [0;1] y0 = 1
x = 1
h = 0, 1
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 7/21
y = −y + x + 1
y = f x + f · f y = 1 + (−y + x + 1)(−1) = y + x
yn+1 = yn + h · (−yn + xn + 1) + h2
2! (yn − xn)2 + E T
y(x)
n = 0 ⇒ x0 = 0 ∧ y0 = 1
y1 = y0 + h · (−y0 + x0 + 1) + h2
2! (y0 − x0)2 = 1, 00500
n = 1 ⇒ x1 = 0, 1 ∧ y1 = 1, 00500
y2 = y1 + h · (−y1 + x1 + 1) + h2
2! (y1 − x1)2 = 1, 01859
n = 9 ⇒ x9 = 0, 9 ∧ y9 = 1, 30101
y10 = y9 + h · (−y9 + x9 + 1) + h2
2! (y9 − x9)2 = 1, 36171
y(1) ≈ 1, 36171
y = f (x, y(x))
y(x0) = y0
h
xn = x0 + n · h
n = 0, 1, 2, 3,... xn ≤ x ≤ xn+1 yn
x = xn
yn+1 = yn +
xn+1
xn
f (t, y(t)) dt
y(t)
y(t)
f
f
x
y
yn+1 = yn + h · f (xn, yn)
yn+1 = yn + h · y(xn, yn) + h2
2! y(ξ )
h2
h
O(h)
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 8/21
y = −y + x + 1
[0;1]
y0 = 1
x = 1 h = 0, 1
f (x, y) = −y + x + 1
yn+1 = yn + h (−yn + xn + 1)
n
yn
n = 0 ⇒ x0 = 0 ∧ y0 = 1
y1 = y0 + h (−y0 + x0 + 1) = 1,0000
n = 1 ⇒ x1 = 0, 1 ∧ y1 = 1
y2 = y1 + h (−y1 + x1 + 1) = 1,0100
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 9/21
n = 9 ⇒ x9 = 0, 9 ∧ y9 = 1, 28742
y10 = y9 + h (−y9 + x9 + 1) = 1,34868
y(1) ≈ 1, 34868
h
h
f (x, y)
xn, yn
mn = f (xn, yn) + f (xn+1, yn+1)
2
y = yn + h
2 ·
f (xn, yn) + f (xn+1, yn+1)
2 · (x − xn)
xn+1 = xn + h
yn+1 = yn + h
2 · [f (xn, yn) + f (xn+1, yn+1)]
xn+1
yn+1 = yn + h
2 [f (xn, yn) + f (xn+1, yn + h · f (xn, yn))]
h3 h2
O(h2)
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 10/21
y = −y + x + 1
[0;1]
y0 = 1
x = 1 h = 0, 1
f (x, y) = −y + x + 1
yn+1 = yn + h
2 [(−yn + xn + 1)
f (xn,yn)
+ (−yn + h (−yn + xn + 1) yn+1 Euler simple
+xn+1 + 1)
n yn
n = 0 ⇒ x0 = 0 , x1 = 0,1 ∧ y0 = 1
y1
= y0
+ h
2 [(−y
0 + x
0 + 1) + (−y
0 + h (−y
0 + x
0 + 1) + x
1 + 1) = 1, 00500
n = 1 ⇒ x1 = 0, 1 , x2 = 0,2 ∧ y1 = 1, 00500
y2 = y1 + h
2 [(−y1 + x1 + 1) + (−y1 + h (−y1 + x1 + 1) + x2 + 1) = 1, 01902
n = 9 ⇒ x0 = 0, 9 , x10 = 1 ∧ y9 = 1, 30723
y10 = y9 + h
2
[(−y9 + x9 + 1) + (−y9 + h (−y9 + x9 + 1) + x10 + 1) = 1, 36854
y(1) ≈ 1, 34854
xn+1
yn+1 = yn + h f (xn, yn)
yk+1n+1 = yn +
h
2 [f (xn, yn) + f (xn+1, ykn+1)]
y = −y + x + 1
[0;1]
y0 = 1
x = 0, 1
10−5
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 11/21
f (x, y) = −y + x + 1
ykn+1 = yn + h (−yn + xn + 1) n = 0, 1, 2, 3,...
k = 0 , n = 0 x0 ∧ y0y01 = y0 + h (−y0 + x0 + 1) = 1, 000000
yk+11 = y0 +
h
2 [f (x0, y0) + f (x1, yk1 )] k = 1, 2, 3,...
E kn = |ykn+1 − yk−1n+1|
k = 1 y11 = y0 + h2
[f (x0, y0) + f (x1, y01)] = 1, 005000 ⇒ E 10 = |y11 − y01 | = 0, 00500
k = 2 y
2
1 = y0 +
h
2 [f (x0, y0) + f (x1, y
1
1)] = 1, 004750 ⇒ E
2
0 = |y
2
1 − y
1
1 | = 0, 00025
k = 3 y31 = y0 + h2
[f (x0, y0) + f (x1, y21)] = 1, 004762 ⇒ E 30 = |y31 − y21 | = 0, 000012
k = 4 y41 = y0 + h2 [f (x0, y0) + f (x1, y31)] = 1, 004763 ⇒ E 40 = |y41 − y31 | < 10−5
n = 1, 2, 3, ..
yn+1 = yn + h φ(xn, yn, h)
φ(xn, yn, h)
[x0; x1]
f (x, y)
yn+1 = yn + w1 K 1 + w2 K 2
K 1 = h f (xn, yn) K 2 = h f (xn + α h , yn + β K 1)
w1, w2, α β
w1 + w2 = 1 w2 = 1
2 α w2 =
1
2 β
α = β = 1
yn+1 = yn + h (1
2K 1 +
1
2K 2)
K 1 = f (xn, yn)K 2 = f (xn + h, yn + h K 1)
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 12/21
α = β = 1/2
yn+1 = yn + h K 2
K 1 = f (xn, yn)
K 2 = f (xn + h2
, yn + h K 12
)
yn+1 = yn + h
6 · (K 1 + 2K 2 + 2K 3 + K 4)
K 1 = f (xn, yn)
K 2 = f (xn + h
2, yn +
h
2K 1)
K 3 = f (xn + h
2, yn +
h
2K 2)
K 4 = f (xn + h, yn + h K 3)
y = −y + x + 1 [0;1] y0 = 1
x = 1
h = 0, 1
K
y(xn)
n = 0 , x0 = 0 ∧ y0 = 1
K 1 = f (x0, y0) = −y0 + x0 + 1 = −1 + 0 + 1 = 0
K 2 = f (x0 + h
2, y0 +
1
2K 1h) = f (0, 05, 1) = −1 + 0, 05 + 1 = 0, 05000
K 3 = f (x0 + h
2, y0 +
1
2K 2h) = f (0, 05, 1, 0025) = −1, 0025 + 0, 05 + 1 = 0, 04750
K 4 = f (x0 + h, y0 + K 3h) = f (0, 1, 1, 00475) = −1, 00475 + 0, 1 + 1 = 0, 09525
y1 = y0 + h
6(K 1 + 2 K 2 + 2 K 3 + k4) = 1, 00484
n = 1 , x1 = 0, 1 ∧ y0 = 1, 00484
K 1 = f (x1, y1) = −y1 + x1 + 1 = −1, 00484 + 0, 1 + 1 = 0, 0951611
K 2 = f (x1 + h
2, y1 +
1
2K 1h) = f (0, 15, 1, 00960) = −1, 00960 + 0, 15 + 1 = 0, 14040
K 3 = f (x1 + h
2, y1 +
1
2K 2h) = f (0, 15, 1, 01186) = −1, 01186 + 0, 15 + 1 = 0, 13814
K 4 = f (x1 + h, y1 + K 3h) = f (0, 2, 1, 01865) = −1, 01865 + 0, 2 + 1 = 0, 018135
y2 = y1 + h
6(K 1 + 2 K 2 + 2 K 3 + k4) = 1, 01873
n = 9 , x9 = 0, 9 ∧ y0 = 1, 30657
y10 = y9 + h6
(K 1 + 2 K 2 + 2 K 3 + k4) = 1, 36788 ⇒ y(1) = 1, 36788
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 13/21
y = f (x, y(x)) y(x0) = y0 h
xn = x0 + n · h
n = 0, 1, 2, 3... y1 y2
y0
y1
y(x) xn
y(xn+h) = yn+1 = yn + h y n + h2
2! yn + ...
y(xn−h) = yn−1 = yn − h y n + h2
2! yn + ...
yn+1 = yn−1 + 2h yn n = 1, 2, 3...
E pT = h3
3 y(ξ )(3)
yn+1
yk+1n+1 = yn +
h
2 [f (xn, yn) + f (xn+1, ykn+1)]
E cT = − h3
12y(ξ )(3)
f (x, y)
N
(xi, yi)
y(x)
[xn−N , xn]
[xn− p, xn+1]
f (x, y)
N = 0 ⇒ yn+1 = yn + h f n
N = 1 ⇒ yn+1 = yn + h
2 (3 f n − f n−1)
N = 2 ⇒ yn+1 = yn + h
12 (23 f n − 16 f n−1 + 5 f n−2)
N = 3 ⇒ yn+1 = yn + h24
(55 f n − 59 f n−1 + 37 f n−2 − 9 f n−3)
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 14/21
y = −y + x + 1
[0;1]
y0 = 1
x = 1 N = 3 y1, y2, y3
h = 0, 1
n xn yn f (xn, yn)
N = 3
y4 = y3 + h
24 (55 f 3 − 59 f 2 + 37 f 1 − 9 f 0) = 1, 07032
y5 = y4 + h
24 (55 f 4 − 59 f 3 + 37 f 2 − 9 f 1) = 1, 10654
y10 = y9 + h
24 (55 f 9 − 59 f 8 + 37 f 7 − 9 f 6) = 1, 36789 ⇒ y(1) = 1, 36789
[xn−N , xn]
[xn− p, xn+1]
yn+1 yn+1
yn+1
f (x, y)
N = 0 ⇒ yn+1 = yn + h
2 · (f n + f n+1)
N = 1 ⇒ yn+1 = yn + h
3(
5
4 f n+1 + 2 f n −
1
4 f n−1)
N = 2 ⇒ yn+1 = yn + h
24(9 f n+1 + 19 f n − 5 f n−1 + f n−2)
N = 3 ⇒ yn+1 = yn + h
720(251 f n+1 + 646 f n − 264 f n−1 + 106 f n−2 − 19 f n−3)
y = −y + x + 1 [0;1] y0 = 1
x = 1
N = 1
y1, y2, y3
h = 0, 1
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 15/21
n xn yn f (xn, yn)
n = 3 y04 = y3 + h
12 (23 f 2 − 16 f 1 + 5 f 0) = 1, 070324
• y14 = y3 + h
3(
5
4 f 04 + 2 f 3 −
1
4 f 2) = 1, 070324
• y24 = y3 + h
3(
5
4 f 14 + 2 f 3 −
1
4 f 2) = 1, 070323
n = 9 y010 = y9 + h12
(23 f 8 − 16 f 7 + 5 f 6) = 1, 367879
• y110 = y9 + h
3(
5
4 f 010 + 2 f 9 −
1
4 f 8) = 1, 367897
• y210 = y9 + h
3(
5
4 f 110 + 2 f 9 −
1
4 f 8) = 1, 367897
⇒ y(1) = 1, 367897
y1 = f 1(x, y1, y2, y3,.....,yN )y2 = f 2(x, y1, y2, y3,.....,yN )y3 = f 3(x, y1, y2, y3,.....,yN )y4 = f 4(x, y1, y2, y3,.....,yN )...........................................yN = f N (x, y1, y2, y3,.....,yN )
y1(x = x0) = y1(0)y2(x = x0) = y2(0)y3(x = x0) = y3(0)y4(x = x0) = y4(0)....................yN (x = x0) = yN (0)
n n = 0, 1, 2, 3, 4,...
y1,n+1 = y1,n + h f 1(x, y1,n, y2,n, y3,n,.....,yN,n) n = 0, 1, 2, 3... y1(x = x0) = y1(0)y2,n+1 = y2,n + h f 2(x, y1,n, y2,n, y3,n,.....,yN,n) n = 0, 1, 2, 3... y2(x = x0) = y2(0)
...........................................yN,n+1 = yN,n + h f N (x, y1,n, y2,n, y3,n,.....,yN,n) n = 0, 1, 2, 3... yN (x = x0) = yN (0)
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 16/21
[ 0 ; 0, 6] h = 0, 2
y1 = f 1(x, y1, y2) = −y1 + 2 x + y2
y2 = f 2(x, y1, y2) = 0, 6 y1 + x y1(x0) = y1,0 = 1, 25
y2(x0) = y2,0 = 2, 34
n = 0 y1,1 = y1,0 + h (−y1,0 + 2 x0 + y2,0) = 1, 46800y2,1 = y2,0 + h (0, 6 y1,0 + x0) = 2, 49000
n = 1 y1,2 = y1,1 + h (−y1,1 + 2 x1 + y2,1) = 1, 75240y2,2 = y2,1 + h (0, 6 y1,1 + x1) = 2, 70616
n = 2 y1,3 = y1,2 + h (−y1,2 + 2 x2 + y2,2) = 2, 10315y2,3 = y2,2 + h (0, 6 y1,2 + x2) = 2, 99645
K 1
N
K 1,1 = f 1(x0, y1,0, y2,0,....,yN,0)K 1,2 = f 2(x0, y1,0, y2,0,....,yN,0)...........................................K 1,N = f N (x0, y1,0, y2,0,....,yN,0)
K 2
K 2,1 = f 1(x0 + h
2, y1,0 +
h
2K 1,1, y2,0 +
h
2K 1,2,....,yN,0 +
h
2K 1,N )
K 2,2 = f 2(x0 + h
2, y1,0 +
h
2K 1,1, y2,0 +
h
2K 1,2,....,yN,0 +
h
2K 1,N )
..................................................................................................
K 2,N = f N (x0 + h
2, y1,0 +
h
2K 1,1, y2,0 +
h
2K 1,2,....,yN,0 +
h
2K 1,N )
K 3
K 3,1 = f 1(x0 + h2
, y1,0 + h2
K 2,1, y2,0 + h2
K 2,2,....,yN,0 + h2
K 2,N )
K 3,2 = f 2(x0 + h
2, y1,0 +
h
2K 2,1, y2,0 +
h
2K 2,2,....,yN,0 +
h
2K 2,N )
..................................................................................................
K 3,N = f N (x0 + h
2, y1,0 +
h
2K 2,1, y2,0 +
h
2K 2,2,....,yN,0 +
h
2K 2,N )
K 3
K 4,1 = f 1(x0 + h, y1,0 + hK 3,1, y2,0 + hK 3,2,....,yN,0 + hK 3,N )K 4,2 = f 2(x0 + h, y1,0 + hK 3,1, y2,0 + hK 3,2,....,yN,0 + hK 3,N )..................................................................................................
K 4,N = f N (x0 + h, y1,0 + hK 3,1, y2,0 + hK 3,2,....,yN,0 + hK 3,N ) x = x1
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 17/21
y1,1 = y1,0 + h
6(K 1,1 + 2 K 2,1 + 2 K 3,1 + K 4,1)
y2,1 = y2,0 + h
6(K 1,2 + 2 K 2,2 + 2 K 3,2 + K 4,2)
...........................................
yN,1 = yN,6 +
h
6 (K 1,N + 2 K 2,N + 2 K 3,N + K 4,N )
y(m) = f (x, y
, y
, y(3),....,y(m−
1))
y(x = x0) = y0 , y(x = x0) = y0 , y(x = x0) = y 0 , y(3)(x = x0) = y(3)0 , ... , y(m)(x = x0) = y(m)
0
y1 = y(x) , y2 = y (x) , y3 = y (x) , y4 = y(3)(x) , ... , ym = y(m−1)(x)
y1 = f 1(x, y1, y2, y3,.....,ym)
y
2 = f 2(x, y1, y2, y3,.....,ym)...................................ym = f m(x, y1, y2, y3,.....,ym)
y1(x = x0) = y0
y2(x = x0) = y
0.............ym(x = x0) = y0
y + 2 y + 5 y = 3 x2 + 10
y(x0) = y0 = 4y(x0) = y 0 = −2
y1 = dy
dx = f 1(x, y1, y2) = y2
y2 = dy2
dx =
d2y2dx2
= y(x) = f 2(x, y1, y2) = 3 x2 + 10 − 5 y1 − 2 y2
y1(x0) = y0 = 4y2(x0) = y 0 = −2
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 18/21
m y(x) + c y(x) + k y(x) = p(x)y1(x = x0) = y0y2(x = x0) = y 0
yn = yn+1 − yn−1
2h yn =
yn+1 − 2yn + yn−1
h2
m yn+1 − 2yn + yn−1
h2 + c
yn+1 − yn−1
2h + k yn = pn
yn+1 =
pn − m
h2
− c
2hyn−1 − k −
2m
h2 yn
m
h2 +
c
2h
n − 1
y n − 1
h
T <
1
π
T = 2π k
m
n
n
x = x0
x = xn
i
y(x) + p(x)y(x) + q (x)y(x) = r(x)
α y(a) + β y(0) = µγ y(b) + δ y(b) = ν
p(x), q (x), r(x) [a, b] α,β,γ,δ,µ,ν
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 19/21
y(x)
y(x) + p(x)y(x) + q (x)y(x) = r(x)
y(x = a) = ya
y(x = b) = yb
[a; b] n h
yi = yi+1 − yi−1
2h
yi = yi+1 − 2yi + yi−1
h2
yi+1 − 2yi + yi−1
h2 + pi
yi+1 − yi−1
2h + q i yi = ri
i 1 −
h
2 pi
yi−1 + (h2 q i − 2) yi +
1 −
h
2 pi
yi+1 = h2 ri i = 1, 2, 3,...n − 1
i = 1 1 − h
2 p
1 y0
+ (h2 q 1
− 2) y1
+ 1 − h
2 p
1 y2
= h2 r1
i = 2
1 −
h
2 p2
y1 + (h2 q 2 − 2) y2 +
1 −
h
2 p2
y3 = h2 r2
i = 3
1 −
h
2 p3
y2 + (h2 q 3 − 2) y3 +
1 −
h
2 p3
y4 = h2 r3
...........................................................................................................
i = n − 1
1 −
h
2 pn−1
yn−2 + (h2 q n−1 − 2) yn−1 +
1 −
h
2 pn−1
yn = h2 rn−1
n − 1 n − 1
y + 4 y = −10 x2
y(x = 0) = 0y(x = 6) = 30
[0; 6]
h = 1
y(x)
yi+1 − 2yi + yi−1
h2 + 4 y
i = −10 x2
i
h = 1
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 20/21
yi−1 + 2 yi + yi+1 = −10 x2i
i = 1 y0 + 2 y1 + y2 = −10 x21
i = 2 y1 + 2 y2 + y3 = −10 x22
i = 3 y2 + 2 y3 + y4 = −10 x23
i = 4 y3 + 2 y4 + y5 = −10 x24
i = 5 y4 + 2 y5 + y6 = −10 x25
2 1 0 0 01 2 1 0 00 1 2 1 0
0 0 1 2 10 0 0 1 2
y1y2y3y4y5
=
−10−40−90
−160−280
⇒
y1y2y3y4y5
=
−2030
−80
40−160
x = 3 y(3) = −80
d
dx( p(x)y(x)) + (q (x) + λ r(x)) y(x) = 0
α y(a) + β y(a) = 0γ y(b) + δ y(b) = 0
p(x) > 0, q (x) > 0
p(x), p(x), q (x), r(x)
[a, b]
α , β , γ , δ
λ
h = 1
y + p y = 0
y(x = 0) = y0 = 0y(x = 4) = y4 = 0
y(x)
yi+1 − 2yi + yi−1
h2 − p yi = 0
h = 1
yi−1 + ( p − 2) yi + yi+1 = 0
i = 1 y0 + ( p − 2) y1 + y2 = 0
i = 2 y1
+ ( p − 2) y2
+ y3
= 0
i = 3 y2 + ( p − 2) y3 + y4 = 0
8/16/2019 AP Ecuaciones Diferenciales Ordinarias
http://slidepdf.com/reader/full/ap-ecuaciones-diferenciales-ordinarias 21/21
(2 − p) −1 0
−1 (2 − p) −10 −1 (2 − p)
y1
y2y3
=
0
00
(2 − p)3 − 2 (2 − p) = 0
p1 = 0,58578 p2 = 2,00000 p3 = 3,41421
p = p1 = 0,58578
1,41422 −1 0−1 1,41422 −10 −1 1,41422
y1y2y3
=
000
y1 = 1
φ1 =
y11
y21y31
=
1,0000
1,41421,0000
p = p2 = 2,00000
φ2 =
y12
y22y32
=
1,0000
0,0000−1,0000
p = p3 = 3,41421
φ3 =
y13
y23y33
=
1,0000
−1,41421,0000