Ejemplos del Metodo de Euler
Ejercicios
En los problemas 1 y 2 utilice el método de
Euler para obtener una aproximación de cuatro cifras del valor que se indica.
Realice a mano la recursión primero con
h = 0.1 y luego con h = 0.05.
1. y ’ = 2x – 3y + 1, y(1) = 5; y(1.2)
Y1 = yo + h*
f(x, y)
h = 0.1
Xo = 1 y = 5 n = 0
Y1 = 5 + 0.1 * (2(1) – 3(5) + 1)
Y1 = 5 -1.2
Y1 = 3.8
Y2 = 3.8 + 0.1 * (2(1.1) – 3(3.8) + 1)
Y2 = 3.8 -0.82
Y2 = 2.9800
h = 0.05
Xo = 1 y = 5 n = 0
Y1 = 5 + 0.05 * (2(1) – 3(5) + 1)
Y1 = 5 -0.6
Y1 = 4.4
Y2 = 4.4 + 0.05 * (2(1.05) – 3(4.4) + 1)
Y2 = 4.4 -0.505
Y2 = 3.895
Y3 = 3.895 + 0.05 * (2(1.1) – 3(3.895) + 1)
Y3 = 3.895 -0.42425
Y3 = 3.47075
Y4 = 3.47075 + 0.05 * (2(1.15) – 3(3.47075) + 1)
Y4 = 3.1151
2. y ’ = x +y ^2,
y(0) = 0; y(0.2)
h = 0.1
Xo = 1 y = 5 n = 0
Y1 = yo + h*
f(x, y)
Y1 = 0 + 0.1 * (0 +0 ^2)
Y1 = 0
Y2 = 0 + 0.1 * (0.1 +0 ^2)
Y2 = 0.01
Y3 = 0.01 + 0.1 * (0.2 + 0.01 ^2)
Y3 = 0.03001
h = 0.05 Xo = 1 y = 5 n = 0
Y1 = yo + h*
f(x, y)
Y1 = 0 + 0.05 * (0 +0 ^2)
Y1 = 0
Y2 = 0 + 0.05 * (0.05 +0 ^2)
Y2 = 2.5 x 10´-3
Y3 = 2.5 x 10 ^-3 + 0.05 * (2.5 x
10 ^-3 +0.05 ^2)
Y3 = 2.75 x
10 ^-3
Y4 = 2.75 x 10 ^-3 + 0.05 * (2.75
x 10 ^-3 +0.1 ^2)
Y4 = 3.3875 x
10 ^-3
Y5 = 3.3875 x 10 ^-3 + 0.05 *
(3.3875 x 10 ^-3 +0.15 ^2)
Y5 = 4.681875
x 10 ^-3
Utilice en los problemas 3 y 4, el método de
Euler para obtener una aproximación de cuatro cifras del valor indicado.
Primero emplee h = 0.1 y después h = 0.005. Encuentre una solución explicita
para cada problema de valor inicial y luego construya tablas similares a las
2.3 y 2.4
3. y ’ = y, y(0) =
1; y(1.0)
h = 0.1
Xo = 0 y = 1 n = 0
Y1 = 1 + 0.1 * (1) = 1.1
Y2 = 1.1 + 0.1 * (1.1) = 1.21
Y3 = 1.21 + 0.1 * (1.21) = 1.331
Y4 = 1.331 + 0.1 * (1.331) = 1.4641
Y5 = 1.4641 + 0.1 * (1.4641) = 1.61051
Y6 = 1.61 + 0.1 * (1.61) = 1.771561
Y7 = 1.77 + 0.1 * (1.77) = 1.9487171
Y8 = 1.9487171 + 0.1 * (1.94897171) = 2.14358881
Y9 = 2.14 + 0.1 * (2.14) = 2.357947691
Y10 = 2.14 + 0.1 * (2.14) = 2.5937
Tabla h = 0.1
Xn
|
Yn
|
0.1
|
1.1
|
0.2
|
1.21
|
0.3
|
1.331
|
0.4
|
1.4641
|
0.5
|
1.61
|
0.6
|
1.77
|
0.7
|
1.94
|
0.8
|
2.14
|
0.9
|
2.35
|
1
|
2.3579
|
h = 0.005
Xo = 0 y = 1 n = 0
y
|
1
|
=
|
1
|
+
|
0.05
|
*
|
1
|
1.05
|
y
|
2
|
=
|
1.05
|
+
|
0.05
|
*
|
1.05
|
1.1025
|
y
|
3
|
=
|
1.1025
|
+
|
0.05
|
*
|
1.1025
|
1.157625
|
y
|
4
|
=
|
1.157625
|
+
|
0.05
|
*
|
1.157625
|
1.21550625
|
y
|
5
|
=
|
1.21550625
|
+
|
0.05
|
*
|
1.21550625
|
1.276281563
|
y
|
6
|
=
|
1.27628156
|
+
|
0.05
|
*
|
1.27628156
|
1.340095641
|
y
|
7
|
=
|
1.34009564
|
+
|
0.05
|
*
|
1.34009564
|
1.407100423
|
y
|
8
|
=
|
1.40710042
|
+
|
0.05
|
*
|
1.40710042
|
1.477455444
|
y
|
9
|
=
|
1.47745544
|
+
|
0.05
|
*
|
1.47745544
|
1.551328216
|
y
|
10
|
=
|
1.55132822
|
+
|
0.05
|
*
|
1.55132822
|
1.628894627
|
y
|
11
|
=
|
1.62889463
|
+
|
0.05
|
*
|
1.62889463
|
1.710339358
|
y
|
12
|
=
|
1.71033936
|
+
|
0.05
|
*
|
1.71033936
|
1.795856326
|
y
|
13
|
=
|
1.79585633
|
+
|
0.05
|
*
|
1.79585633
|
1.885649142
|
y
|
14
|
=
|
1.88564914
|
+
|
0.05
|
*
|
1.88564914
|
1.979931599
|
y
|
15
|
=
|
1.9799316
|
+
|
0.05
|
*
|
1.9799316
|
2.078928179
|
y
|
16
|
=
|
2.07892818
|
+
|
0.05
|
*
|
2.07892818
|
2.182874588
|
y
|
17
|
=
|
2.18287459
|
+
|
0.05
|
*
|
2.18287459
|
2.292018318
|
y
|
18
|
=
|
2.29201832
|
+
|
0.05
|
*
|
2.29201832
|
2.406619234
|
y
|
19
|
=
|
2.40661923
|
+
|
0.05
|
*
|
2.40661923
|
2.526950195
|
y
|
20
|
=
|
2.5269502
|
+
|
0.05
|
*
|
2.5269502
|
2.653297705
|
h = 0.1
Xo = 0 y = 1 n = 0
4. y ’ = 2xy, y(1) = 1; y(1.5)
h = 0.1 Xo = 1 y = 1 n = 0
Yo
|
=
|
Y
|
+
|
n(0.1)
|
*
|
2
|
*
|
x
|
*
|
y
|
=
|
Y(n+1)
|
y1
|
=
|
1
|
+
|
0.1
|
*
|
2
|
*
|
1
|
*
|
1
|
=
|
1.2
|
y2
|
=
|
1.2
|
+
|
0.1
|
*
|
2
|
*
|
1.1
|
*
|
1.2
|
=
|
1.464
|
y3
|
=
|
1.464
|
+
|
0.1
|
*
|
2
|
*
|
1.2
|
*
|
1.464
|
=
|
1.8154
|
y4
|
=
|
1.81536
|
+
|
0.1
|
*
|
2
|
*
|
1.3
|
*
|
1.8154
|
=
|
2.2874
|
y5
|
=
|
2.28735
|
+
|
0.1
|
*
|
2
|
*
|
1.4
|
*
|
2.2874
|
=
|
2.9278
|
y6
|
=
|
2.92781
|
+
|
0.1
|
*
|
2
|
*
|
1.5
|
*
|
2.9278
|
=
|
3.8062
|
y7
|
=
|
3.80616
|
+
|
0.1
|
*
|
2
|
*
|
1.6
|
*
|
3.8062
|
=
|
5.0241
|
y8
|
=
|
5.02413
|
+
|
0.1
|
*
|
2
|
*
|
1.7
|
*
|
5.0241
|
=
|
6.7323
|
y9
|
=
|
6.73233
|
+
|
0.1
|
*
|
2
|
*
|
1.8
|
*
|
6.7323
|
=
|
9.156
|
y10
|
=
|
9.15597
|
+
|
0.1
|
*
|
2
|
*
|
1.9
|
*
|
9.156
|
=
|
12.635
|
y11
|
=
|
12.6352
|
+
|
0.1
|
*
|
2
|
*
|
2
|
*
|
12.635
|
=
|
17.689
|
y12
|
=
|
17.6893
|
+
|
0.1
|
*
|
2
|
*
|
2.1
|
*
|
17.689
|
=
|
25.119
|
y13
|
=
|
25.1188
|
+
|
0.1
|
*
|
2
|
*
|
2.2
|
*
|
25.119
|
=
|
36.171
|
y14
|
=
|
36.1711
|
+
|
0.1
|
*
|
2
|
*
|
2.3
|
*
|
36.171
|
=
|
52.81
|
y15
|
=
|
52.8099
|
+
|
0.1
|
*
|
2
|
*
|
2.4
|
*
|
52.81
|
=
|
78.159
|
Tabla
xn
|
yn
|
0.1
|
1.2
|
0.2
|
1.464
|
0.3
|
1.81536
|
0.4
|
2.2873536
|
0.5
|
2.92781261
|
0.6
|
3.80615639
|
0.7
|
5.02412644
|
0.8
|
6.73232942
|
0.9
|
9.15596802
|
1
|
12.6352359
|
1.1
|
17.6893302
|
1.2
|
25.1188489
|
1.3
|
36.1711424
|
1.4
|
52.8098679
|
1.5
|
78.1586045
|
h = 0.05
Xo = 0 y = 1 n = 0
Yo
|
=
|
Y
|
+
|
n(0.05)
|
*
|
2
|
*
|
x
|
*
|
y
|
=
|
Y(n+1)
|
y1
|
=
|
1
|
+
|
0.05
|
*
|
2
|
*
|
1
|
*
|
1
|
=
|
1.1
|
y2
|
=
|
1.1
|
+
|
0.05
|
*
|
2
|
*
|
1.05
|
*
|
1.1
|
=
|
1.2155
|
y3
|
=
|
1.2155
|
+
|
0.05
|
*
|
2
|
*
|
1.1
|
*
|
1.2155
|
=
|
1.3492
|
y4
|
=
|
1.34921
|
+
|
0.05
|
*
|
2
|
*
|
1.15
|
*
|
1.3492
|
=
|
1.5044
|
y5
|
=
|
1.50436
|
+
|
0.05
|
*
|
2
|
*
|
1.2
|
*
|
1.5044
|
=
|
1.6849
|
y6
|
=
|
1.68489
|
+
|
0.05
|
*
|
2
|
*
|
1.25
|
*
|
1.6849
|
=
|
1.8955
|
y7
|
=
|
1.8955
|
+
|
0.05
|
*
|
2
|
*
|
1.3
|
*
|
1.8955
|
=
|
2.1419
|
y8
|
=
|
2.14191
|
+
|
0.05
|
*
|
2
|
*
|
1.35
|
*
|
2.1419
|
=
|
2.4311
|
y9
|
=
|
2.43107
|
+
|
0.05
|
*
|
2
|
*
|
1.4
|
*
|
2.4311
|
=
|
2.7714
|
y10
|
=
|
2.77142
|
+
|
0.05
|
*
|
2
|
*
|
1.45
|
*
|
2.7714
|
=
|
3.1733
|
y11
|
=
|
3.17328
|
+
|
0.05
|
*
|
2
|
*
|
1.5
|
*
|
3.1733
|
=
|
3.6493
|
y12
|
=
|
3.64927
|
+
|
0.05
|
*
|
2
|
*
|
1.55
|
*
|
3.6493
|
=
|
4.2149
|
y13
|
=
|
4.21491
|
+
|
0.05
|
*
|
2
|
*
|
1.6
|
*
|
4.2149
|
=
|
4.8893
|
y14
|
=
|
4.88929
|
+
|
0.05
|
*
|
2
|
*
|
1.65
|
*
|
4.8893
|
=
|
5.696
|
y15
|
=
|
5.69602
|
+
|
0.05
|
*
|
2
|
*
|
1.7
|
*
|
5.696
|
=
|
6.6643
|
y16
|
=
|
6.66435
|
+
|
0.05
|
*
|
2
|
*
|
1.75
|
6.6643
|
=
|
7.8306
|
|
y17
|
=
|
7.83061
|
+
|
0.05
|
*
|
2
|
*
|
1.8
|
7.8306
|
=
|
9.2401
|
|
y18
|
=
|
9.24012
|
+
|
0.05
|
*
|
2
|
*
|
1.85
|
9.2401
|
=
|
10.95
|
|
y19
|
=
|
10.9495
|
+
|
0.05
|
*
|
2
|
*
|
1.9
|
10.95
|
=
|
13.03
|
|
y20
|
=
|
13.03
|
+
|
0.05
|
*
|
2
|
*
|
1.95
|
13.03
|
=
|
15.571
|
|
y21
|
=
|
15.5708
|
+
|
0.05
|
*
|
2
|
*
|
2
|
15.571
|
=
|
18.685
|
|
y22
|
=
|
18.6849
|
+
|
0.05
|
*
|
2
|
*
|
2.05
|
18.685
|
=
|
22.515
|
|
y23
|
=
|
22.5154
|
+
|
0.05
|
*
|
2
|
*
|
2.1
|
22.515
|
=
|
27.244
|
|
y24
|
=
|
27.2436
|
+
|
0.05
|
*
|
2
|
*
|
2.15
|
27.244
|
=
|
33.101
|
|
y25
|
=
|
33.101
|
+
|
0.05
|
*
|
2
|
*
|
2.2
|
33.101
|
=
|
40.383
|
|
y26
|
=
|
40.3832
|
+
|
0.05
|
*
|
2
|
*
|
2.25
|
40.383
|
=
|
49.469
|
|
y27
|
=
|
49.4694
|
+
|
0.05
|
*
|
2
|
*
|
2.3
|
49.469
|
=
|
60.847
|
|
y28
|
=
|
60.8473
|
+
|
0.05
|
*
|
2
|
*
|
2.35
|
60.847
|
=
|
75.146
|
|
y29
|
=
|
75.1465
|
+
|
0.05
|
*
|
2
|
*
|
2.4
|
75.146
|
=
|
93.182
|
Tabla xn yn
0.1
|
1.1
|
0.15
|
1.2155
|
0.2
|
1.349205
|
0.25
|
1.50436358
|
0.3
|
1.6848872
|
0.35
|
1.8954981
|
0.4
|
2.14191286
|
0.45
|
2.43107109
|
0.5
|
2.77142105
|
0.55
|
3.1732771
|
0.6
|
3.64926866
|
0.65
|
4.21490531
|
0.7
|
4.88929016
|
0.75
|
5.69602303
|
0.8
|
6.66434695
|
0.85
|
7.83060766
|
0.9
|
9.24011704
|
0.95
|
10.9495387
|
1
|
13.029951
|
1.05
|
15.5707915
|
1.1
|
18.6849498
|
1.15
|
22.5153645
|
1.2
|
27.2435911
|
1.25
|
33.1009631
|
1.3
|
40.383175
|
1.35
|
49.4693894
|
1.4
|
60.847349
|
1.45
|
75.146476
|
1.5
|
93.1816302
|
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