Ejemplos del Metodo de Euler.

• BY:Carlos Dario Flores
• 05/07/2014

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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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