Tarea II. Circuitos Eléctricos. Propedeútico UMSNH

Tarea Ii: Ecuaciones diferenciales ordinarias; mtodos de runge kutta

Universidad Michoacana de San Nicols de Hidalgo Teora de Control.25/Noviembre/2014Alumno: Ramos Albarrn Fernando

Universidad Michoacana de San Nicols de Hidalgo

Asesor: Dr. Jaime Cerda Jacobo.

Curso Propedutico: Mtodos Numricos.

[Tarea Ii: Ecuaciones diferenciales ordinarias; mtodos de runge kutta] Ingeniera Elctrica

Se modific el cdigo original para aplicar el mtodo por separado, para ejecutar un script (exp_sin_b_2.m) por separado llamando a las funciones originales, en ste caso el mtodo de Euler, cmo su funcin es la primera en ser llamada, es en el que se grafica la funcin original, ese cdigo se omite en las funciones de los dems mtodos. Los archivos .m por separado, estn programados para proporcionar la grfica del mtodo en comparacin con la funcin original, que se introduzca su derivada, cada funcin recibe dicha funcin y calcula su primitiva para ser comparada; adems se proporcionan los valores en forma tabulada.

Nota: En los archivos .m originales, hay vario cdigo en comentarios, la mayora es significativo y sin gran relevancia, ya que se fue perfeccionando poco a poco hasta corregir errores de programacin y se omite en ste documento. Hay otro cdigo que se puede usar para proporcionar valores que se omiten en las tabulaciones. Al final de ste texto, se agregan los cdigos de las funciones utilizadas.

Cdigos del Script para la funcin:

Para b=2

syms x f;f='exp(-x)*(2*cos(2*x)-sin(2*x))';ti=0;tf=2*pi;Xi=0;Yi=0;h=0.3;Euler_Method(f,ti,tf,Xi,Yi,h);Heun_Method(f,ti,tf,Xi,Yi,h);Mid_Method(f,ti,tf,Xi,Yi,h);Runge_Kutta2_Ralston(f,ti,tf,Xi,Yi,h);Runge_Kutta3(f,ti,tf,Xi,Yi,h);Runge_Kutta4(f,ti,tf,Xi,Yi,h);title('Comparacin de los mtodos de Runge Kutta con la funcin original');xlabel('Tiempo t (seg)');ylabel('f(t)');legend('Euler','Solucin verdadera','Heun','Punto Medio','Ralston','Runge Kutta 3', 'Runge Kutta4');

Salida en la ventana de comandos

primitiva = sin(2*x)*exp(-x)

i x Y_Verdadero Y_Euler Pendiente Y+1_Euler

B =

0 0 0 0 0 0 1.0000 0.3000 0.4183 0.6000 2.0000 0.6000 2.0000 0.6000 0.5115 0.8414 0.8045 0.8414 3.0000 0.9000 0.3959 0.8072 -0.1138 0.8072 4.0000 1.2000 0.2034 0.6330 -0.5807 0.6330 5.0000 1.5000 0.0315 0.4387 -0.6476 0.4387 6.0000 1.8000 -0.0731 0.2967 -0.4733 0.2967 7.0000 2.1000 -0.1067 0.2298 -0.2233 0.2298 8.0000 2.4000 -0.0904 0.2257 -0.0133 0.2257 9.0000 2.7000 -0.0519 0.2576 0.1062 0.2576 10.0000 3.0000 -0.0139 0.2988 0.1372 0.2988 11.0000 3.3000 0.0115 0.3317 0.1095 0.3317 12.0000 3.6000 0.0217 0.3492 0.0586 0.3492 13.0000 3.9000 0.0202 0.3527 0.0116 0.3527 14.0000 4.2000 0.0128 0.3473 -0.0180 0.3473 15.0000 4.5000 0.0046 0.3388 -0.0284 0.3388 16.0000 4.8000 -0.0014 0.3313 -0.0248 0.3313 17.0000 5.1000 -0.0043 0.3269 -0.0148 0.3269 18.0000 5.4000 -0.0044 0.3256 -0.0044 0.3256 19.0000 5.7000 -0.0031 0.3264 0.0027 0.3264 20.0000 6.0000 -0.0013 0.3281 0.0057 0.3281

x Y_Verdadero Y_Heun

B =

0 0 0 0.3000 0.4183 0.4207 0.6000 0.5115 0.5243 0.9000 0.3959 0.4201 1.2000 0.2034 0.2359 1.5000 0.0315 0.0677 1.8000 -0.0731 -0.0368 2.1000 -0.1067 -0.0722 2.4000 -0.0904 -0.0583 2.7000 -0.0519 -0.0218 3.0000 -0.0139 0.0152 3.3000 0.0115 0.0404 3.6000 0.0217 0.0510 3.9000 0.0202 0.0500 4.2000 0.0128 0.0430 4.5000 0.0046 0.0351 4.8000 -0.0014 0.0291 5.1000 -0.0043 0.0262 5.4000 -0.0044 0.0260 5.7000 -0.0031 0.0272 6.0000 -0.0013 0.0289

x Y_Verdadero Y_Punto_Medio

B =

0 0 0 0.3000 0.4183 0.4171 0.6000 0.5115 0.5050 0.9000 0.3959 0.3837 1.2000 0.2034 0.1871 1.5000 0.0315 0.0132 1.8000 -0.0731 -0.0915 2.1000 -0.1067 -0.1241 2.4000 -0.0904 -0.1065 2.7000 -0.0519 -0.0671 3.0000 -0.0139 -0.0286 3.3000 0.0115 -0.0031 3.6000 0.0217 0.0069 3.9000 0.0202 0.0052 4.2000 0.0128 -0.0024 4.5000 0.0046 -0.0108 4.8000 -0.0014 -0.0168 5.1000 -0.0043 -0.0196 5.4000 -0.0044 -0.0197 5.7000 -0.0031 -0.0183 6.0000 -0.0013 -0.0166

x Y_Verdadero Y_Runge_Kutta2_Ralston

B =

0 0 0 0.3000 0.4183 0.4181 0.6000 0.5115 0.5137 0.9000 0.3959 0.4011 1.2000 0.2034 0.2109 1.5000 0.0315 0.0402 1.8000 -0.0731 -0.0642 2.1000 -0.1067 -0.0982 2.4000 -0.0904 -0.0825 2.7000 -0.0519 -0.0446 3.0000 -0.0139 -0.0068 3.3000 0.0115 0.0185 3.6000 0.0217 0.0287 3.9000 0.0202 0.0274 4.2000 0.0128 0.0201 4.5000 0.0046 0.0119 4.8000 -0.0014 0.0059 5.1000 -0.0043 0.0031 5.4000 -0.0044 0.0029 5.7000 -0.0031 0.0042 6.0000 -0.0013 0.0060

x Y_Verdadero Y_Runge_Kutta3

B =

0 0 0 0.3000 0.4183 0.4475 0.6000 0.5115 0.5611 0.9000 0.3959 0.4543 1.2000 0.2034 0.2614 1.5000 0.0315 0.0842 1.8000 -0.0731 -0.0268 2.1000 -0.1067 -0.0651 2.4000 -0.0904 -0.0511 2.7000 -0.0519 -0.0130 3.0000 -0.0139 0.0260 3.3000 0.0115 0.0527 3.6000 0.0217 0.0640 3.9000 0.0202 0.0631 4.2000 0.0128 0.0559 4.5000 0.0046 0.0475 4.8000 -0.0014 0.0412 5.1000 -0.0043 0.0381 5.4000 -0.0044 0.0378 5.7000 -0.0031 0.0391 6.0000 -0.0013 0.0409


B =

0 1.0000 0 0.3000 1.4183 0.4475 0.6000 1.5115 0.5611 0.9000 1.3959 0.4543 1.2000 1.2034 0.2614 1.5000 1.0315 0.0842 1.8000 0.9269 -0.0268 2.1000 0.8933 -0.0651 2.4000 0.9096 -0.0511 2.7000 0.9481 -0.0130 3.0000 0.9861 0.0260 3.3000 1.0115 0.0527 3.6000 1.0217 0.0640 3.9000 1.0202 0.0631 4.2000 1.0128 0.0559 4.5000 1.0046 0.0475 4.8000 0.9986 0.0412 5.1000 0.9957 0.0381 5.4000 0.9956 0.0378 5.7000 0.9969 0.0391 6.0000 0.9987 0.0409

A partir de aqu slo se mostrarn los valores de entrada para el script dnde se llama a las funciones, y las salidas del programa para los valores restantes de b, esto, para no ser tan repetitivo. Se omite tambin la salida de la primitiva, ya que es de la misma forma, para diferentes valores de b.

Para b=4

syms x f;f='exp(-x)*(4*cos(4*x)-sin(4*x))';ti=0;tf=2*pi;Xi=0;Yi=0;h=0.1;



B =

0 0 0 0 0 0 1.0000 0.1000 0.3524 0.4000 4.0000 0.4000 2.0000 0.2000 0.5873 0.6981 2.9813 0.6981 3.0000 0.3000 0.6905 0.8676 1.6943 0.8676 4.0000 0.4000 0.6700 0.9059 0.3833 0.9059 5.0000 0.5000 0.5515 0.8311 -0.7483 0.8311 6.0000 0.6000 0.3707 0.6749 -1.5611 0.6749 7.0000 0.7000 0.1664 0.4760 -1.9895 0.4760 8.0000 0.8000 -0.0262 0.2722 -2.0379 0.2722 9.0000 0.9000 -0.1799 0.0954 -1.7680 0.0954 10.0000 1.0000 -0.2784 -0.0324 -1.2785 -0.0324 11.0000 1.1000 -0.3168 -0.1008 -0.6834 -0.1008 12.0000 1.2000 -0.3000 -0.1100 -0.0924 -0.1100 13.0000 1.3000 -0.2408 -0.0695 0.4055 -0.0695 14.0000 1.4000 -0.1557 0.0057 0.7515 0.0057 15.0000 1.5000 -0.0623 0.0977 0.9207 0.0977 16.0000 1.6000 0.0235 0.1897 0.9193 0.1897 17.0000 1.7000 0.0903 0.2675 0.7786 0.2675 18.0000 1.8000 0.1312 0.3220 0.5450 0.3220 19.0000 1.9000 0.1448 0.3491 0.2710 0.3491 20.0000 2.0000 0.1339 0.3497 0.0056 0.3497 21.0000 2.1000 0.1047 0.3284 -0.2127 0.3284 22.0000 2.2000 0.0648 0.2925 -0.3590 0.2925 23.0000 2.3000 0.0223 0.2501 -0.4243 0.2501 24.0000 2.4000 -0.0158 0.2088 -0.4133 0.2088 25.0000 2.5000 -0.0447 0.1746 -0.3415 0.1746 26.0000 2.6000 -0.0615 0.1515 -0.2308 0.1515 27.0000 2.7000 -0.0659 0.1410 -0.1052 0.1410 28.0000 2.8000 -0.0595 0.1424 0.0137 0.1424 29.0000 2.9000 -0.0453 0.1533 0.1089 0.1533 30.0000 3.0000 -0.0267 0.1703 0.1704 0.1703 31.0000 3.1000 -0.0075 0.1898 0.1948 0.1898 32.0000 3.2000 0.0094 0.2083 0.1852 0.2083 33.0000 3.3000 0.0218 0.2232 0.1492 0.2232 34.0000 3.4000 0.0287 0.2329 0.0971 0.2329 35.0000 3.5000 0.0299 0.2369 0.0396 0.2369 36.0000 3.6000 0.0264 0.2355 -0.0134 0.2355 37.0000 3.7000 0.0195 0.2301 -0.0548 0.2301 38.0000 3.8000 0.0109 0.2220 -0.0803 0.2220 39.0000 3.9000 0.0022 0.2131 -0.0891 0.2131 40.0000 4.0000 -0.0053 0.2049 -0.0827 0.2049 41.0000 4.1000 -0.0106 0.1984 -0.0649 0.1984 42.0000 4.2000 -0.0133 0.1943 -0.0405 0.1943 43.0000 4.3000 -0.0135 0.1929 -0.0143 0.1929 44.0000 4.4000 -0.0116 0.1938 0.0093 0.1938 45.0000 4.5000 -0.0083 0.1965 0.0272 0.1965 46.0000 4.6000 -0.0044 0.2003 0.0377 0.2003 47.0000 4.7000 -0.0005 0.2044 0.0406 0.2044 48.0000 4.8000 0.0028 0.2080 0.0368 0.2080 49.0000 4.9000 0.0051 0.2108 0.0281 0.2108 50.0000 5.0000 0.0062 0.2125 0.0167 0.2125 51.0000 5.1000 0.0061 0.2130 0.0048 0.2130 52.0000 5.2000 0.0051 0.2124 -0.0056 0.2124 53.0000 5.3000 0.0035 0.2111 -0.0133 0.2111 54.0000 5.4000 0.0017 0.2093 -0.0176 0.2093 55.0000 5.5000 -0.0000 0.2075 -0.0184 0.2075 56.0000 5.6000 -0.0015 0.2059 -0.0163 0.2059 57.0000 5.7000 -0.0024 0.2047 -0.0121 0.2047 58.0000 5.8000 -0.0028 0.2040 -0.0068 0.2040 59.0000 5.9000 -0.0027 0.2038 -0.0015 0.2038 60.0000 6.0000 -0.0022 0.2042 0.0032 0.2042 61.0000 6.1000 -0.0015 0.2048 0.0065 0.2048 62.0000 6.2000 -0.0007 0.2056 0.0082 0.2056


B =

0 0 0 0.1000 0.3524 0.3491 0.2000 0.5873 0.5828 0.3000 0.6905 0.6867 0.4000 0.6700 0.6685 0.5000 0.5515 0.5530 0.6000 0.3707 0.3755 0.7000 0.1664 0.1741 0.8000 -0.0262 -0.0162 0.9000 -0.1799 -0.1685 1.0000 -0.2784 -0.2666 1.1000 -0.3168 -0.3054 1.2000 -0.3000 -0.2898 1.3000 -0.2408 -0.2319 1.4000 -0.1557 -0.1483 1.5000 -0.0623 -0.0563 1.6000 0.0235 0.0286 1.7000 0.0903 0.0948 1.8000 0.1312 0.1356 1.9000 0.1448 0.1494 2.0000 0.1339 0.1391 2.1000 0.1047 0.1105 2.2000 0.0648 0.0713 2.3000 0.0223 0.0294 2.4000 -0.0158 -0.0083 2.5000 -0.0447 -0.0369 2.6000 -0.0615 -0.0537 2.7000 -0.0659 -0.0583 2.8000 -0.0595 -0.0522 2.9000 -0.0453 -0.0382 3.0000 -0.0267 -0.0200 3.1000 -0.0075 -0.0010 3.2000 0.0094 0.0158 3.3000 0.0218 0.0281 3.4000 0.0287 0.0349 3.5000 0.0299 0.0362 3.6000 0.0264 0.0328 3.7000 0.0195 0.0260 3.8000 0.0109 0.0176 3.9000 0.0022 0.0090 4.0000 -0.0053 0.0016 4.1000 -0.0106 -0.0037 4.2000 -0.0133 -0.0064 4.3000 -0.0135 -0.0066 4.4000 -0.0116 -0.0048 4.5000 -0.0083 -0.0016 4.6000 -0.0044 0.0023 4.7000 -0.0005 0.0062 4.8000 0.0028 0.0094 4.9000 0.0051 0.0117 5.0000 0.0062 0.0128 5.1000 0.0061 0.0127 5.2000 0.0051 0.0118 5.3000 0.0035 0.0102 5.4000 0.0017 0.0084 5.5000 -0.0000 0.0067 5.6000 -0.0015 0.0053 5.7000 -0.0024 0.0043 5.8000 -0.0028 0.0039 5.9000 -0.0027 0.0040 6.0000 -0.0022 0.0045 6.1000 -0.0015 0.0052 6.2000 -0.0007 0.0060


B =

0 0 0 0.1000 0.3524 0.3540 0.2000 0.5873 0.5896 0.3000 0.6905 0.6923 0.4000 0.6700 0.6708 0.5000 0.5515 0.5508 0.6000 0.3707 0.3683 0.7000 0.1664 0.1625 0.8000 -0.0262 -0.0313 0.9000 -0.1799 -0.1856 1.0000 -0.2784 -0.2843 1.1000 -0.3168 -0.3225 1.2000 -0.3000 -0.3052 1.3000 -0.2408 -0.2452 1.4000 -0.1557 -0.1594 1.5000 -0.0623 -0.0654 1.6000 0.0235 0.0210 1.7000 0.0903 0.0880 1.8000 0.1312 0.1290 1.9000 0.1448 0.1424 2.0000 0.1339 0.1313 2.1000 0.1047 0.1017 2.2000 0.0648 0.0616 2.3000 0.0223 0.0188 2.4000 -0.0158 -0.0196 2.5000 -0.0447 -0.0485 2.6000 -0.0615 -0.0654 2.7000 -0.0659 -0.0697 2.8000 -0.0595 -0.0632 2.9000 -0.0453 -0.0488 3.0000 -0.0267 -0.0301 3.1000 -0.0075 -0.0107 3.2000 0.0094 0.0063 3.3000 0.0218 0.0187 3.4000 0.0287 0.0255 3.5000 0.0299 0.0268 3.6000 0.0264 0.0232 3.7000 0.0195 0.0162 3.8000 0.0109 0.0075 3.9000 0.0022 -0.0012 4.0000 -0.0053 -0.0087 4.1000 -0.0106 -0.0140 4.2000 -0.0133 -0.0168 4.3000 -0.0135 -0.0170 4.4000 -0.0116 -0.0151 4.5000 -0.0083 -0.0117 4.6000 -0.0044 -0.0077 4.7000 -0.0005 -0.0038 4.8000 0.0028 -0.0005 4.9000 0.0051 0.0018 5.0000 0.0062 0.0028 5.1000 0.0061 0.0028 5.2000 0.0051 0.0018 5.3000 0.0035 0.0002 5.4000 0.0017 -0.0016 5.5000 -0.0000 -0.0034 5.6000 -0.0015 -0.0049 5.7000 -0.0024 -0.0058 5.8000 -0.0028 -0.0062 5.9000 -0.0027 -0.0061 6.0000 -0.0022 -0.0056 6.1000 -0.0015 -0.0049 6.2000 -0.0007 -0.0040


B =

0 0 0 0.1000 0.3524 0.3514 0.2000 0.5873 0.5859 0.3000 0.6905 0.6892 0.4000 0.6700 0.6692 0.5000 0.5515 0.5514 0.6000 0.3707 0.3714 0.7000 0.1664 0.1679 0.8000 -0.0262 -0.0241 0.9000 -0.1799 -0.1774 1.0000 -0.2784 -0.2757 1.1000 -0.3168 -0.3141 1.2000 -0.3000 -0.2977 1.3000 -0.2408 -0.2387 1.4000 -0.1557 -0.1540 1.5000 -0.0623 -0.0610 1.6000 0.0235 0.0246 1.7000 0.0903 0.0911 1.8000 0.1312 0.1320 1.9000 0.1448 0.1456 2.0000 0.1339 0.1349 2.1000 0.1047 0.1058 2.2000 0.0648 0.0661 2.3000 0.0223 0.0238 2.4000 -0.0158 -0.0142 2.5000 -0.0447 -0.0430 2.6000 -0.0615 -0.0598 2.7000 -0.0659 -0.0643 2.8000 -0.0595 -0.0580 2.9000 -0.0453 -0.0438 3.0000 -0.0267 -0.0253 3.1000 -0.0075 -0.0061 3.2000 0.0094 0.0108 3.3000 0.0218 0.0231 3.4000 0.0287 0.0300 3.5000 0.0299 0.0312 3.6000 0.0264 0.0277 3.7000 0.0195 0.0208 3.8000 0.0109 0.0123 3.9000 0.0022 0.0036 4.0000 -0.0053 -0.0038 4.1000 -0.0106 -0.0091 4.2000 -0.0133 -0.0118 4.3000 -0.0135 -0.0121 4.4000 -0.0116 -0.0102 4.5000 -0.0083 -0.0069 4.6000 -0.0044 -0.0030 4.7000 -0.0005 0.0009 4.8000 0.0028 0.0042 4.9000 0.0051 0.0065 5.0000 0.0062 0.0075 5.1000 0.0061 0.0075 5.2000 0.0051 0.0065 5.3000 0.0035 0.0049 5.4000 0.0017 0.0031 5.5000 -0.0000 0.0014 5.6000 -0.0015 -0.0001 5.7000 -0.0024 -0.0010 5.8000 -0.0028 -0.0014 5.9000 -0.0027 -0.0013 6.0000 -0.0022 -0.0008 6.1000 -0.0015 -0.0001 6.2000 -0.0007 0.0007


B =

0 0 0 0.1000 0.3524 0.3617 0.2000 0.5873 0.6077 0.3000 0.6905 0.7215 0.4000 0.6700 0.7100 0.5000 0.5515 0.5975 0.6000 0.3707 0.4194 0.7000 0.1664 0.2147 0.8000 -0.0262 0.0193 0.9000 -0.1799 -0.1388 1.0000 -0.2784 -0.2423 1.1000 -0.3168 -0.2855 1.2000 -0.3000 -0.2727 1.3000 -0.2408 -0.2159 1.4000 -0.1557 -0.1319 1.5000 -0.0623 -0.0382 1.6000 0.0235 0.0491 1.7000 0.0903 0.1179 1.8000 0.1312 0.1612 1.9000 0.1448 0.1769 2.0000 0.1339 0.1677 2.1000 0.1047 0.1395 2.2000 0.0648 0.1000 2.3000 0.0223 0.0573 2.4000 -0.0158 0.0185 2.5000 -0.0447 -0.0113 2.6000 -0.0615 -0.0292 2.7000 -0.0659 -0.0346 2.8000 -0.0595 -0.0290 2.9000 -0.0453 -0.0151 3.0000 -0.0267 0.0033 3.1000 -0.0075 0.0227 3.2000 0.0094 0.0399 3.3000 0.0218 0.0528 3.4000 0.0287 0.0601 3.5000 0.0299 0.0618 3.6000 0.0264 0.0586 3.7000 0.0195 0.0518 3.8000 0.0109 0.0433 3.9000 0.0022 0.0345 4.0000 -0.0053 0.0269 4.1000 -0.0106 0.0214 4.2000 -0.0133 0.0184 4.3000 -0.0135 0.0180 4.4000 -0.0116 0.0197 4.5000 -0.0083 0.0230 4.6000 -0.0044 0.0269 4.7000 -0.0005 0.0309 4.8000 0.0028 0.0343 4.9000 0.0051 0.0366 5.0000 0.0062 0.0378 5.1000 0.0061 0.0378 5.2000 0.0051 0.0369 5.3000 0.0035 0.0353 5.4000 0.0017 0.0335 5.5000 -0.0000 0.0317 5.6000 -0.0015 0.0303 5.7000 -0.0024 0.0293 5.8000 -0.0028 0.0288 5.9000 -0.0027 0.0289 6.0000 -0.0022 0.0293 6.1000 -0.0015 0.0301 6.2000 -0.0007 0.0309


B =

0 1.0000 0 0.1000 1.3524 0.3617 0.2000 1.5873 0.6077 0.3000 1.6905 0.7215 0.4000 1.6700 0.7100 0.5000 1.5515 0.5975 0.6000 1.3707 0.4194 0.7000 1.1664 0.2147 0.8000 0.9738 0.0193 0.9000 0.8201 -0.1388 1.0000 0.7216 -0.2423 1.1000 0.6832 -0.2855 1.2000 0.7000 -0.2727 1.3000 0.7592 -0.2159 1.4000 0.8443 -0.1319 1.5000 0.9377 -0.0382 1.6000 1.0235 0.0491 1.7000 1.0903 0.1179 1.8000 1.1312 0.1612 1.9000 1.1448 0.1769 2.0000 1.1339 0.1677 2.1000 1.1047 0.1395 2.2000 1.0648 0.1000 2.3000 1.0223 0.0573 2.4000 0.9842 0.0185 2.5000 0.9553 -0.0113 2.6000 0.9385 -0.0292 2.7000 0.9341 -0.0346 2.8000 0.9405 -0.0290 2.9000 0.9547 -0.0151 3.0000 0.9733 0.0033 3.1000 0.9925 0.0227 3.2000 1.0094 0.0399 3.3000 1.0218 0.0528 3.4000 1.0287 0.0601 3.5000 1.0299 0.0618 3.6000 1.0264 0.0586 3.7000 1.0195 0.0518 3.8000 1.0109 0.0433 3.9000 1.0022 0.0345 4.0000 0.9947 0.0269 4.1000 0.9894 0.0214 4.2000 0.9867 0.0184 4.3000 0.9865 0.0180 4.4000 0.9884 0.0197 4.5000 0.9917 0.0230 4.6000 0.9956 0.0269 4.7000 0.9995 0.0309 4.8000 1.0028 0.0343 4.9000 1.0051 0.0366 5.0000 1.0062 0.0378 5.1000 1.0061 0.0378 5.2000 1.0051 0.0369 5.3000 1.0035 0.0353 5.4000 1.0017 0.0335 5.5000 1.0000 0.0317 5.6000 0.9985 0.0303 5.7000 0.9976 0.0293 5.8000 0.9972 0.0288 5.9000 0.9973 0.0289 6.0000 0.9978 0.0293 6.1000 0.9985 0.0301 6.2000 0.9993 0.0309

Para b=6




B =

0 0 0 0 0 0 1.0000 0.1000 0.5109 0.6000 6.0000 0.6000 2.0000 0.2000 0.7631 0.9970 3.9699 0.9970 3.0000 0.3000 0.7214 1.0987 1.0170 1.0987 4.0000 0.4000 0.4528 0.9255 -1.7313 0.9255 5.0000 0.5000 0.0856 0.5837 -3.4185 0.5837 6.0000 0.6000 -0.2429 0.2149 -3.6884 0.2149 7.0000 0.7000 -0.4328 -0.0561 -2.7100 -0.0561 8.0000 0.8000 -0.4476 -0.1589 -1.0279 -0.1589 9.0000 0.9000 -0.3142 -0.0906 0.6835 -0.0906 10.0000 1.0000 -0.1028 0.0957 1.8625 0.0957 11.0000 1.1000 0.1037 0.3179 2.2222 0.3179 12.0000 1.2000 0.2390 0.4973 1.7941 0.4973 13.0000 1.3000 0.2721 0.5833 0.8603 0.5833 14.0000 1.4000 0.2107 0.5649 -0.1839 0.5649 15.0000 1.5000 0.0920 0.4670 -0.9791 0.4670 16.0000 1.6000 -0.0352 0.3358 -1.3118 0.3358 17.0000 1.7000 -0.1279 0.2201 -1.1576 0.2201 18.0000 1.8000 -0.1621 0.1546 -0.6551 0.1546 19.0000 1.9000 -0.1375 0.1515 -0.0306 0.1515 20.0000 2.0000 -0.0726 0.2006 0.4906 0.2006 21.0000 2.1000 0.0041 0.2764 0.7578 0.2764 22.0000 2.2000 0.0656 0.3494 0.7302 0.3494 23.0000 2.3000 0.0946 0.3964 0.4702 0.3964 24.0000 2.4000 0.0876 0.4068 0.1044 0.4068 25.0000 2.5000 0.0534 0.3839 -0.2290 0.3839 26.0000 2.6000 0.0080 0.3412 -0.4275 0.3412 27.0000 2.7000 -0.0317 0.2961 -0.4511 0.2961 28.0000 2.8000 -0.0540 0.2637 -0.3236 0.2637 29.0000 2.9000 -0.0546 0.2523 -0.1141 0.2523 30.0000 3.0000 -0.0374 0.2618 0.0945 0.2618 31.0000 3.1000 -0.0111 0.2852 0.2346 0.2852 32.0000 3.2000 0.0140 0.3125 0.2730 0.3125 33.0000 3.3000 0.0300 0.3341 0.2157 0.3341 34.0000 3.4000 0.0334 0.3440 0.0986 0.3440 35.0000 3.5000 0.0253 0.3410 -0.0293 0.3410 36.0000 3.6000 0.0104 0.3286 -0.1245 0.3286 37.0000 3.7000 -0.0051 0.3124 -0.1620 0.3124 38.0000 3.8000 -0.0162 0.2984 -0.1400 0.2984 39.0000 3.9000 -0.0200 0.2907 -0.0765 0.2907 40.0000 4.0000 -0.0166 0.2908 0.0004 0.2908 41.0000 4.1000 -0.0084 0.2971 0.0632 0.2971 42.0000 4.2000 0.0010 0.3065 0.0941 0.3065 43.0000 4.3000 0.0084 0.3154 0.0888 0.3154 44.0000 4.4000 0.0117 0.3209 0.0556 0.3209 45.0000 4.5000 0.0106 0.3220 0.0103 0.3220 46.0000 4.6000 0.0063 0.3190 -0.0301 0.3190 47.0000 4.7000 0.0007 0.3136 -0.0534 0.3136 48.0000 4.8000 -0.0041 0.3081 -0.0551 0.3081 49.0000 4.9000 -0.0067 0.3043 -0.0386 0.3043 50.0000 5.0000 -0.0067 0.3030 -0.0125 0.3030 51.0000 5.1000 -0.0044 0.3043 0.0129 0.3043 52.0000 5.2000 -0.0012 0.3072 0.0295 0.3072 53.0000 5.3000 0.0019 0.3106 0.0335 0.3106 54.0000 5.4000 0.0038 0.3132 0.0259 0.3132 55.0000 5.5000 0.0041 0.3143 0.0112 0.3143 56.0000 5.6000 0.0030 0.3139 -0.0044 0.3139 57.0000 5.7000 0.0012 0.3123 -0.0158 0.3123 58.0000 5.8000 -0.0007 0.3103 -0.0200 0.3103 59.0000 5.9000 -0.0020 0.3086 -0.0169 0.3086 60.0000 6.0000 -0.0025 0.3077 -0.0089 0.3077 61.0000 6.1000 -0.0020 0.3078 0.0006 0.3078 62.0000 6.2000 -0.0010 0.3086 0.0081 0.3086


B =

0 0 0 0.1000 0.5109 0.4985 0.2000 0.7631 0.7478 0.3000 0.7214 0.7121 0.4000 0.4528 0.4546 0.5000 0.0856 0.0993 0.6000 -0.2429 -0.2206 0.7000 -0.4328 -0.4075 0.8000 -0.4476 -0.4248 0.9000 -0.3142 -0.2975 1.0000 -0.1028 -0.0932 1.1000 0.1037 0.1076 1.2000 0.2390 0.2403 1.3000 0.2721 0.2741 1.4000 0.2107 0.2160 1.5000 0.0920 0.1014 1.6000 -0.0352 -0.0220 1.7000 -0.1279 -0.1127 1.8000 -0.1621 -0.1470 1.9000 -0.1375 -0.1239 2.0000 -0.0726 -0.0615 2.1000 0.0041 0.0129 2.2000 0.0656 0.0729 2.3000 0.0946 0.1016 2.4000 0.0876 0.0954 2.5000 0.0534 0.0626 2.6000 0.0080 0.0186 2.7000 -0.0317 -0.0201 2.8000 -0.0540 -0.0420 2.9000 -0.0546 -0.0430 3.0000 -0.0374 -0.0265 3.1000 -0.0111 -0.0011 3.2000 0.0140 0.0233 3.3000 0.0300 0.0390 3.4000 0.0334 0.0425 3.5000 0.0253 0.0348 3.6000 0.0104 0.0205 3.7000 -0.0051 0.0054 3.8000 -0.0162 -0.0054 3.9000 -0.0200 -0.0092 4.0000 -0.0166 -0.0061 4.1000 -0.0084 0.0018 4.2000 0.0010 0.0109 4.3000 0.0084 0.0182 4.4000 0.0117 0.0215 4.5000 0.0106 0.0205 4.6000 0.0063 0.0163 4.7000 0.0007 0.0109 4.8000 -0.0041 0.0062 4.9000 -0.0067 0.0036 5.0000 -0.0067 0.0036 5.1000 -0.0044 0.0058 5.2000 -0.0012 0.0089 5.3000 0.0019 0.0119 5.4000 0.0038 0.0137 5.5000 0.0041 0.0141 5.6000 0.0030 0.0131 5.7000 0.0012 0.0113 5.8000 -0.0007 0.0094 5.9000 -0.0020 0.0082 6.0000 -0.0025 0.0077 6.1000 -0.0020 0.0082 6.2000 -0.0010 0.0092


B =

0 0 0 0.1000 0.5109 0.5171 0.2000 0.7631 0.7707 0.3000 0.7214 0.7261 0.4000 0.4528 0.4518 0.5000 0.0856 0.0787 0.6000 -0.2429 -0.2540 0.7000 -0.4328 -0.4455 0.8000 -0.4476 -0.4591 0.9000 -0.3142 -0.3226 1.0000 -0.1028 -0.1076 1.1000 0.1037 0.1017 1.2000 0.2390 0.2384 1.3000 0.2721 0.2711 1.4000 0.2107 0.2081 1.5000 0.0920 0.0872 1.6000 -0.0352 -0.0418 1.7000 -0.1279 -0.1355 1.8000 -0.1621 -0.1698 1.9000 -0.1375 -0.1443 2.0000 -0.0726 -0.0782 2.1000 0.0041 -0.0003 2.2000 0.0656 0.0619 2.3000 0.0946 0.0911 2.4000 0.0876 0.0837 2.5000 0.0534 0.0487 2.6000 0.0080 0.0026 2.7000 -0.0317 -0.0376 2.8000 -0.0540 -0.0600 2.9000 -0.0546 -0.0605 3.0000 -0.0374 -0.0429 3.1000 -0.0111 -0.0162 3.2000 0.0140 0.0093 3.3000 0.0300 0.0255 3.4000 0.0334 0.0288 3.5000 0.0253 0.0204 3.6000 0.0104 0.0053 3.7000 -0.0051 -0.0104 3.8000 -0.0162 -0.0216 3.9000 -0.0200 -0.0254 4.0000 -0.0166 -0.0219 4.1000 -0.0084 -0.0136 4.2000 0.0010 -0.0040 4.3000 0.0084 0.0035 4.4000 0.0117 0.0068 4.5000 0.0106 0.0057 4.6000 0.0063 0.0012 4.7000 0.0007 -0.0045 4.8000 -0.0041 -0.0093 4.9000 -0.0067 -0.0119 5.0000 -0.0067 -0.0119 5.1000 -0.0044 -0.0096 5.2000 -0.0012 -0.0063 5.3000 0.0019 -0.0032 5.4000 0.0038 -0.0013 5.5000 0.0041 -0.0010 5.6000 0.0030 -0.0020 5.7000 0.0012 -0.0039 5.8000 -0.0007 -0.0059 5.9000 -0.0020 -0.0072 6.0000 -0.0025 -0.0076 6.1000 -0.0020 -0.0071 6.2000 -0.0010 -0.0061


B =

0 0 0 0.1000 0.5109 0.5073 0.2000 0.7631 0.7581 0.3000 0.7214 0.7175 0.4000 0.4528 0.4514 0.5000 0.0856 0.0873 0.6000 -0.2429 -0.2387 0.7000 -0.4328 -0.4275 0.8000 -0.4476 -0.4426 0.9000 -0.3142 -0.3106 1.0000 -0.1028 -0.1010 1.1000 0.1037 0.1039 1.2000 0.2390 0.2384 1.3000 0.2721 0.2715 1.4000 0.2107 0.2108 1.5000 0.0920 0.0930 1.6000 -0.0352 -0.0331 1.7000 -0.1279 -0.1252 1.8000 -0.1621 -0.1593 1.9000 -0.1375 -0.1350 2.0000 -0.0726 -0.0707 2.1000 0.0041 0.0054 2.2000 0.0656 0.0664 2.3000 0.0946 0.0953 2.4000 0.0876 0.0884 2.5000 0.0534 0.0545 2.6000 0.0080 0.0095 2.7000 -0.0317 -0.0299 2.8000 -0.0540 -0.0520 2.9000 -0.0546 -0.0527 3.0000 -0.0374 -0.0356 3.1000 -0.0111 -0.0096 3.2000 0.0140 0.0153 3.3000 0.0300 0.0312 3.4000 0.0334 0.0346 3.5000 0.0253 0.0266 3.6000 0.0104 0.0119 3.7000 -0.0051 -0.0036 3.8000 -0.0162 -0.0145 3.9000 -0.0200 -0.0183 4.0000 -0.0166 -0.0150 4.1000 -0.0084 -0.0069 4.2000 0.0010 0.0025 4.3000 0.0084 0.0098 4.4000 0.0117 0.0131 4.5000 0.0106 0.0120 4.6000 0.0063 0.0077 4.7000 0.0007 0.0022 4.8000 -0.0041 -0.0026 4.9000 -0.0067 -0.0052 5.0000 -0.0067 -0.0051 5.1000 -0.0044 -0.0029 5.2000 -0.0012 0.0003 5.3000 0.0019 0.0033 5.4000 0.0038 0.0052 5.5000 0.0041 0.0056 5.6000 0.0030 0.0045 5.7000 0.0012 0.0027 5.8000 -0.0007 0.0008 5.9000 -0.0020 -0.0005 6.0000 -0.0025 -0.0009 6.1000 -0.0020 -0.0005 6.2000 -0.0010 0.0005


B =

0 0 0 0.1000 0.5109 0.5309 0.2000 0.7631 0.8084 0.3000 0.7214 0.7882 0.4000 0.4528 0.5308 0.5000 0.0856 0.1629 0.6000 -0.2429 -0.1759 0.7000 -0.4328 -0.3806 0.8000 -0.4476 -0.4091 0.9000 -0.3142 -0.2839 1.0000 -0.1028 -0.0737 1.1000 0.1037 0.1378 1.2000 0.2390 0.2815 1.3000 0.2721 0.3231 1.4000 0.2107 0.2676 1.5000 0.0920 0.1505 1.6000 -0.0352 0.0211 1.7000 -0.1279 -0.0762 1.8000 -0.1621 -0.1157 1.9000 -0.1375 -0.0950 2.0000 -0.0726 -0.0317 2.1000 0.0041 0.0458 2.2000 0.0656 0.1098 2.3000 0.0946 0.1420 2.4000 0.0876 0.1375 2.5000 0.0534 0.1046 2.6000 0.0080 0.0591 2.7000 -0.0317 0.0180 2.8000 -0.0540 -0.0061 2.9000 -0.0546 -0.0084 3.0000 -0.0374 0.0079 3.1000 -0.0111 0.0341 3.2000 0.0140 0.0598 3.3000 0.0300 0.0769 3.4000 0.0334 0.0813 3.5000 0.0253 0.0739 3.6000 0.0104 0.0592 3.7000 -0.0051 0.0434 3.8000 -0.0162 0.0317 3.9000 -0.0200 0.0273 4.0000 -0.0166 0.0302 4.1000 -0.0084 0.0382 4.2000 0.0010 0.0477 4.3000 0.0084 0.0555 4.4000 0.0117 0.0592 4.5000 0.0106 0.0584 4.6000 0.0063 0.0542 4.7000 0.0007 0.0485 4.8000 -0.0041 0.0436 4.9000 -0.0067 0.0408 5.0000 -0.0067 0.0406 5.1000 -0.0044 0.0427 5.2000 -0.0012 0.0460 5.3000 0.0019 0.0491 5.4000 0.0038 0.0511 5.5000 0.0041 0.0516 5.6000 0.0030 0.0506 5.7000 0.0012 0.0488 5.8000 -0.0007 0.0468 5.9000 -0.0020 0.0454 6.0000 -0.0025 0.0450 6.1000 -0.0020 0.0454 6.2000 -0.0010 0.0464


B =

0 1.0000 0 0.1000 1.5109 0.5309 0.2000 1.7631 0.8084 0.3000 1.7214 0.7882 0.4000 1.4528 0.5308 0.5000 1.0856 0.1629 0.6000 0.7571 -0.1759 0.7000 0.5672 -0.3806 0.8000 0.5524 -0.4091 0.9000 0.6858 -0.2839 1.0000 0.8972 -0.0737 1.1000 1.1037 0.1378 1.2000 1.2390 0.2815 1.3000 1.2721 0.3231 1.4000 1.2107 0.2676 1.5000 1.0920 0.1505 1.6000 0.9648 0.0211 1.7000 0.8721 -0.0762 1.8000 0.8379 -0.1157 1.9000 0.8625 -0.0950 2.0000 0.9274 -0.0317 2.1000 1.0041 0.0458 2.2000 1.0656 0.1098 2.3000 1.0946 0.1420 2.4000 1.0876 0.1375 2.5000 1.0534 0.1046 2.6000 1.0080 0.0591 2.7000 0.9683 0.0180 2.8000 0.9460 -0.0061 2.9000 0.9454 -0.0084 3.0000 0.9626 0.0079 3.1000 0.9889 0.0341 3.2000 1.0140 0.0598 3.3000 1.0300 0.0769 3.4000 1.0334 0.0813 3.5000 1.0253 0.0739 3.6000 1.0104 0.0592 3.7000 0.9949 0.0434 3.8000 0.9838 0.0317 3.9000 0.9800 0.0273 4.0000 0.9834 0.0302 4.1000 0.9916 0.0382 4.2000 1.0010 0.0477 4.3000 1.0084 0.0555 4.4000 1.0117 0.0592 4.5000 1.0106 0.0584 4.6000 1.0063 0.0542 4.7000 1.0007 0.0485 4.8000 0.9959 0.0436 4.9000 0.9933 0.0408 5.0000 0.9933 0.0406 5.1000 0.9956 0.0427 5.2000 0.9988 0.0460 5.3000 1.0019 0.0491 5.4000 1.0038 0.0511 5.5000 1.0041 0.0516 5.6000 1.0030 0.0506 5.7000 1.0012 0.0488 5.8000 0.9993 0.0468 5.9000 0.9980 0.0454 6.0000 0.9975 0.0450 6.1000 0.9980 0.0454 6.2000 0.9990 0.0464

Para b=8




B =

0 0 0 0 0 0 1.0000 0.1000 0.6491 0.8000 8.0000 0.8000 2.0000 0.2000 0.8184 1.2394 4.3942 1.2394 3.0000 0.3000 0.5004 1.1385 -1.0096 1.1385 4.0000 0.4000 -0.0391 0.6514 -4.8706 0.6514 5.0000 0.5000 -0.4590 0.1200 -5.3143 0.1200 6.0000 0.6000 -0.5467 -0.1513 -2.7126 -0.1513 7.0000 0.7000 -0.3135 -0.0582 0.9309 -0.0582 8.0000 0.8000 0.0524 0.2812 3.3946 0.2812 9.0000 0.9000 0.3227 0.6330 3.5178 0.6330 10.0000 1.0000 0.3640 0.7986 1.6560 0.7986 11.0000 1.1000 0.1947 0.7194 -0.7922 0.7194 12.0000 1.2000 -0.0525 0.4839 -2.3546 0.4839 13.0000 1.3000 -0.2256 0.2519 -2.3202 0.2519 14.0000 1.4000 -0.2415 0.1522 -0.9975 0.1522 15.0000 1.5000 -0.1197 0.2164 0.6419 0.2164 16.0000 1.6000 0.0467 0.3790 1.6260 0.3790 17.0000 1.7000 0.1570 0.5314 1.5246 0.5314 18.0000 1.8000 0.1596 0.5905 0.5909 0.5905 19.0000 1.9000 0.0727 0.5402 -0.5032 0.5402 20.0000 2.0000 -0.0390 0.4284 -1.1182 0.4284 21.0000 2.1000 -0.1087 0.3286 -0.9979 0.3286 22.0000 2.2000 -0.1051 0.2943 -0.3426 0.2943 23.0000 2.3000 -0.0436 0.3328 0.3850 0.3328 24.0000 2.4000 0.0311 0.4094 0.7659 0.4094 25.0000 2.5000 0.0749 0.4745 0.6505 0.4745 26.0000 2.6000 0.0690 0.4938 0.1930 0.4938 27.0000 2.7000 0.0256 0.4649 -0.2892 0.4649 28.0000 2.8000 -0.0242 0.4126 -0.5227 0.4126 29.0000 2.9000 -0.0515 0.3704 -0.4222 0.3704 30.0000 3.0000 -0.0451 0.3599 -0.1044 0.3599 31.0000 3.1000 -0.0147 0.3813 0.2140 0.3813 32.0000 3.2000 0.0184 0.4169 0.3553 0.4169 33.0000 3.3000 0.0352 0.4442 0.2728 0.4442 34.0000 3.4000 0.0293 0.4495 0.0530 0.4495 35.0000 3.5000 0.0082 0.4338 -0.1565 0.4338 36.0000 3.6000 -0.0137 0.4097 -0.2407 0.4097 37.0000 3.7000 -0.0240 0.3922 -0.1754 0.3922 38.0000 3.8000 -0.0190 0.3898 -0.0240 0.3898 39.0000 3.9000 -0.0043 0.4011 0.1133 0.4011 40.0000 4.0000 0.0101 0.4174 0.1625 0.4174 41.0000 4.1000 0.0163 0.4286 0.1121 0.4286 42.0000 4.2000 0.0123 0.4294 0.0083 0.4294 43.0000 4.3000 0.0021 0.4213 -0.0813 0.4213 44.0000 4.4000 -0.0074 0.4104 -0.1093 0.4104 45.0000 4.5000 -0.0110 0.4032 -0.0713 0.4032 46.0000 4.6000 -0.0079 0.4032 -0.0004 0.4032 47.0000 4.7000 -0.0009 0.4090 0.0579 0.4090 48.0000 4.8000 0.0053 0.4163 0.0733 0.4163 49.0000 4.9000 0.0074 0.4208 0.0450 0.4208 50.0000 5.0000 0.0050 0.4205 -0.0033 0.4205 51.0000 5.1000 0.0002 0.4164 -0.0410 0.4164 52.0000 5.2000 -0.0038 0.4115 -0.0490 0.4115 53.0000 5.3000 -0.0050 0.4087 -0.0282 0.4087 54.0000 5.4000 -0.0032 0.4091 0.0045 0.4091 55.0000 5.5000 0.0001 0.4120 0.0288 0.4120 56.0000 5.6000 0.0027 0.4153 0.0326 0.4153 57.0000 5.7000 0.0033 0.4170 0.0175 0.4170 58.0000 5.8000 0.0020 0.4166 -0.0046 0.4166 59.0000 5.9000 -0.0002 0.4145 -0.0202 0.4145 60.0000 6.0000 -0.0019 0.4124 -0.0216 0.4124 61.0000 6.1000 -0.0022 0.4113 -0.0108 0.4113 62.0000 6.2000 -0.0013 0.4117 0.0041 0.4117


B =

0 0 0 0.1000 0.6491 0.6197 0.2000 0.8184 0.7889 0.3000 0.5004 0.4949 0.4000 -0.0391 -0.0143 0.5000 -0.4590 -0.4157 0.6000 -0.5467 -0.5048 0.7000 -0.3135 -0.2885 0.8000 0.0524 0.0571 0.9000 0.3227 0.3158 1.0000 0.3640 0.3590 1.1000 0.1947 0.2017 1.2000 -0.0525 -0.0321 1.3000 -0.2256 -0.1979 1.4000 -0.2415 -0.2157 1.5000 -0.1197 -0.1023 1.6000 0.0467 0.0552 1.7000 0.1570 0.1610 1.8000 0.1596 0.1654 1.9000 0.0727 0.0843 2.0000 -0.0390 -0.0215 2.1000 -0.1087 -0.0885 2.2000 -0.1051 -0.0864 2.3000 -0.0436 -0.0289 2.4000 0.0311 0.0420 2.5000 0.0749 0.0841 2.6000 0.0690 0.0793 2.7000 0.0256 0.0387 2.8000 -0.0242 -0.0085 2.9000 -0.0515 -0.0348 3.0000 -0.0451 -0.0294 3.1000 -0.0147 -0.0009 3.2000 0.0184 0.0305 3.3000 0.0352 0.0468 3.4000 0.0293 0.0416 3.5000 0.0082 0.0218 3.6000 -0.0137 0.0010 3.7000 -0.0240 -0.0090 3.8000 -0.0190 -0.0045 3.9000 -0.0043 0.0092 4.0000 0.0101 0.0230 4.1000 0.0163 0.0290 4.2000 0.0123 0.0253 4.3000 0.0021 0.0158 4.4000 -0.0074 0.0068 4.5000 -0.0110 0.0032 4.6000 -0.0079 0.0061 4.7000 -0.0009 0.0126 4.8000 0.0053 0.0186 4.9000 0.0074 0.0206 5.0000 0.0050 0.0184 5.1000 0.0002 0.0139 5.2000 -0.0038 0.0101 5.3000 -0.0050 0.0089 5.4000 -0.0032 0.0106 5.5000 0.0001 0.0136 5.6000 0.0027 0.0161 5.7000 0.0033 0.0168 5.8000 0.0020 0.0155 5.9000 -0.0002 0.0135 6.0000 -0.0019 0.0118 6.1000 -0.0022 0.0115 6.2000 -0.0013 0.0124


B =

0 0 0 0.1000 0.6491 0.6639 0.2000 0.8184 0.8332 0.3000 0.5004 0.5031 0.4000 -0.0391 -0.0517 0.5000 -0.4590 -0.4809 0.6000 -0.5467 -0.5679 0.7000 -0.3135 -0.3261 0.8000 0.0524 0.0499 0.9000 0.3227 0.3261 1.0000 0.3640 0.3664 1.1000 0.1947 0.1911 1.2000 -0.0525 -0.0629 1.3000 -0.2256 -0.2396 1.4000 -0.2415 -0.2545 1.5000 -0.1197 -0.1285 1.6000 0.0467 0.0424 1.7000 0.1570 0.1549 1.8000 0.1596 0.1567 1.9000 0.0727 0.0669 2.0000 -0.0390 -0.0478 2.1000 -0.1087 -0.1189 2.2000 -0.1051 -0.1146 2.3000 -0.0436 -0.0510 2.4000 0.0311 0.0256 2.5000 0.0749 0.0703 2.6000 0.0690 0.0637 2.7000 0.0256 0.0190 2.8000 -0.0242 -0.0321 2.9000 -0.0515 -0.0599 3.0000 -0.0451 -0.0531 3.1000 -0.0147 -0.0217 3.2000 0.0184 0.0122 3.3000 0.0352 0.0293 3.4000 0.0293 0.0231 3.5000 0.0082 0.0013 3.6000 -0.0137 -0.0212 3.7000 -0.0240 -0.0316 3.8000 -0.0190 -0.0264 3.9000 -0.0043 -0.0112 4.0000 0.0101 0.0036 4.1000 0.0163 0.0098 4.2000 0.0123 0.0056 4.3000 0.0021 -0.0048 4.4000 -0.0074 -0.0145 4.5000 -0.0110 -0.0182 4.6000 -0.0079 -0.0150 4.7000 -0.0009 -0.0078 4.8000 0.0053 -0.0014 4.9000 0.0074 0.0007 5.0000 0.0050 -0.0018 5.1000 0.0002 -0.0067 5.2000 -0.0038 -0.0108 5.3000 -0.0050 -0.0120 5.4000 -0.0032 -0.0102 5.5000 0.0001 -0.0068 5.6000 0.0027 -0.0041 5.7000 0.0033 -0.0035 5.8000 0.0020 -0.0049 5.9000 -0.0002 -0.0072 6.0000 -0.0019 -0.0089 6.1000 -0.0022 -0.0092 6.2000 -0.0013 -0.0082


B =

0 0 0 0.1000 0.6491 0.6401 0.2000 0.8184 0.8075 0.3000 0.5004 0.4945 0.4000 -0.0391 -0.0372 0.5000 -0.4590 -0.4513 0.6000 -0.5467 -0.5381 0.7000 -0.3135 -0.3085 0.8000 0.0524 0.0521 0.9000 0.3227 0.3187 1.0000 0.3640 0.3596 1.1000 0.1947 0.1930 1.2000 -0.0525 -0.0507 1.3000 -0.2256 -0.2214 1.4000 -0.2415 -0.2372 1.5000 -0.1197 -0.1173 1.6000 0.0467 0.0467 1.7000 0.1570 0.1555 1.8000 0.1596 0.1582 1.9000 0.0727 0.0727 2.0000 -0.0390 -0.0375 2.1000 -0.1087 -0.1062 2.2000 -0.1051 -0.1028 2.3000 -0.0436 -0.0422 2.4000 0.0311 0.0315 2.5000 0.0749 0.0747 2.6000 0.0690 0.0689 2.7000 0.0256 0.0261 2.8000 -0.0242 -0.0229 2.9000 -0.0515 -0.0499 3.0000 -0.0451 -0.0436 3.1000 -0.0147 -0.0137 3.2000 0.0184 0.0189 3.3000 0.0352 0.0355 3.4000 0.0293 0.0298 3.5000 0.0082 0.0089 3.6000 -0.0137 -0.0127 3.7000 -0.0240 -0.0228 3.8000 -0.0190 -0.0179 3.9000 -0.0043 -0.0035 4.0000 0.0101 0.0108 4.1000 0.0163 0.0169 4.2000 0.0123 0.0129 4.3000 0.0021 0.0029 4.4000 -0.0074 -0.0064 4.5000 -0.0110 -0.0100 4.6000 -0.0079 -0.0069 4.7000 -0.0009 -0.0001 4.8000 0.0053 0.0061 4.9000 0.0074 0.0082 5.0000 0.0050 0.0058 5.1000 0.0002 0.0011 5.2000 -0.0038 -0.0029 5.3000 -0.0050 -0.0041 5.4000 -0.0032 -0.0023 5.5000 0.0001 0.0009 5.6000 0.0027 0.0035 5.7000 0.0033 0.0041 5.8000 0.0020 0.0028 5.9000 -0.0002 0.0006 6.0000 -0.0019 -0.0010 6.1000 -0.0022 -0.0014 6.2000 -0.0013 -0.0004


B =

0 0 0 0.1000 0.6491 0.6866 0.2000 0.8184 0.9009 0.3000 0.5004 0.6090 0.4000 -0.0391 0.0655 0.5000 -0.4590 -0.3808 0.6000 -0.5467 -0.4985 0.7000 -0.3135 -0.2815 0.8000 0.0524 0.0885 0.9000 0.3227 0.3772 1.0000 0.3640 0.4384 1.1000 0.1947 0.2792 1.2000 -0.0525 0.0283 1.3000 -0.2256 -0.1577 1.4000 -0.2415 -0.1867 1.5000 -0.1197 -0.0711 1.6000 0.0467 0.0985 1.7000 0.1570 0.2176 1.8000 0.1596 0.2290 1.9000 0.0727 0.1458 2.0000 -0.0390 0.0315 2.1000 -0.1087 -0.0443 2.2000 -0.1051 -0.0465 2.3000 -0.0436 0.0130 2.4000 0.0311 0.0896 2.5000 0.0749 0.1376 2.6000 0.0690 0.1354 2.7000 0.0256 0.0933 2.8000 -0.0242 0.0420 2.9000 -0.0515 0.0118 3.0000 -0.0451 0.0158 3.1000 -0.0147 0.0454 3.2000 0.0184 0.0796 3.3000 0.0352 0.0984 3.4000 0.0293 0.0942 3.5000 0.0082 0.0734 3.6000 -0.0137 0.0507 3.7000 -0.0240 0.0390 3.8000 -0.0190 0.0430 3.9000 -0.0043 0.0575 4.0000 0.0101 0.0725 4.1000 0.0163 0.0796 4.2000 0.0123 0.0762 4.3000 0.0021 0.0662 4.4000 -0.0074 0.0563 4.5000 -0.0110 0.0520 4.6000 -0.0079 0.0547 4.7000 -0.0009 0.0617 4.8000 0.0053 0.0682 4.9000 0.0074 0.0707 5.0000 0.0050 0.0686 5.1000 0.0002 0.0638 5.2000 -0.0038 0.0595 5.3000 -0.0050 0.0581 5.4000 -0.0032 0.0597 5.5000 0.0001 0.0630 5.6000 0.0027 0.0658 5.7000 0.0033 0.0666 5.8000 0.0020 0.0654 5.9000 -0.0002 0.0631 6.0000 -0.0019 0.0613 6.1000 -0.0022 0.0609 6.2000 -0.0013 0.0618


B =

0 1.0000 0 0.1000 1.6491 0.6866 0.2000 1.8184 0.9009 0.3000 1.5004 0.6090 0.4000 0.9609 0.0655 0.5000 0.5410 -0.3808 0.6000 0.4533 -0.4985 0.7000 0.6865 -0.2815 0.8000 1.0524 0.0885 0.9000 1.3227 0.3772 1.0000 1.3640 0.4384 1.1000 1.1947 0.2792 1.2000 0.9475 0.0283 1.3000 0.7744 -0.1577 1.4000 0.7585 -0.1867 1.5000 0.8803 -0.0711 1.6000 1.0467 0.0985 1.7000 1.1570 0.2176 1.8000 1.1596 0.2290 1.9000 1.0727 0.1458 2.0000 0.9610 0.0315 2.1000 0.8913 -0.0443 2.2000 0.8949 -0.0465 2.3000 0.9564 0.0130 2.4000 1.0311 0.0896 2.5000 1.0749 0.1376 2.6000 1.0690 0.1354 2.7000 1.0256 0.0933 2.8000 0.9758 0.0420 2.9000 0.9485 0.0118 3.0000 0.9549 0.0158 3.1000 0.9853 0.0454 3.2000 1.0184 0.0796 3.3000 1.0352 0.0984 3.4000 1.0293 0.0942 3.5000 1.0082 0.0734 3.6000 0.9863 0.0507 3.7000 0.9760 0.0390 3.8000 0.9810 0.0430 3.9000 0.9957 0.0575 4.0000 1.0101 0.0725 4.1000 1.0163 0.0796 4.2000 1.0123 0.0762 4.3000 1.0021 0.0662 4.4000 0.9926 0.0563 4.5000 0.9890 0.0520 4.6000 0.9921 0.0547 4.7000 0.9991 0.0617 4.8000 1.0053 0.0682 4.9000 1.0074 0.0707 5.0000 1.0050 0.0686 5.1000 1.0002 0.0638 5.2000 0.9962 0.0595 5.3000 0.9950 0.0581 5.4000 0.9968 0.0597 5.5000 1.0001 0.0630 5.6000 1.0027 0.0658 5.7000 1.0033 0.0666 5.8000 1.0020 0.0654 5.9000 0.9998 0.0631 6.0000 0.9981 0.0613 6.1000 0.9978 0.0609 6.2000 0.9987 0.0618

Cdigo para el Mtodo de Euler:

function Euler_Method(f,ti,tf,Xi,Yi,h)n=(tf-ti)/h;x1=zeros(n,1);y1=zeros(n,1);x1=[ti:h:tf];y1(1)=Yi;y2(1)=Yi;A(1)=0;for k=1:n x=x1(k); y=y1(k); fp(k+1)=eval(f); y2(k+1)=y1(k)+(fp(k+1))*h; y1(k+1)=y2(k+1); A(k+1,1)=k;endplot(x1,y1,'-r')primitiva=int(sym(f))Q=length(x1);for u=1:Q x=x1(u); y3(u)=eval(primitiva);endhold onplot(x1,y3,'-b')disp(' i x Y_Verdadero Y_Euler Pendiente Y+1_Euler')B=[A, x1', y3', y1, fp', y2']end

Cdigo para el Mtodo de Heun:

function Heun_Method(f,ti,tf,Xi,Yi,h)n=(tf-ti)/h;x1=zeros(n,1);y1=zeros(n,1);x1=[ti:h:tf];y1(1)=Yi;y2(1)=Yi;A(1)=0;for k=1:n x=x1(k); y=y1(k); fp(k+1)=eval(f); y2(k+1)=y1(k)+(fp(k+1))*h; y=y2(k+1); x=x1(k+1); fp2(k+1)=eval(f); feval(k+1)=(fp(k+1)+fp2(k+1))/2; y1(k+1)=y1(k)+feval(k+1)*h; A(k+1,1)=k;endplot(x1,y1,'-g')primitiva=int(sym(f));Q=length(x1);for u=1:Q x=x1(u); y4(u)=eval(primitiva);enddisp(' x Y_Verdadero Y_Heun')B=[x1', y4', y1] end

Para b=10




B =

0 0 0 0 0 0 1.0000 0.1000 0.7614 1.0000 10.0000 1.0000 2.0000 0.2000 0.7445 1.4127 4.1275 1.4127 3.0000 0.3000 0.1045 0.9976 -4.1516 0.9976 4.0000 0.4000 -0.5073 0.2537 -7.4386 0.2537 5.0000 0.5000 -0.5816 -0.1337 -3.8742 -0.1337 6.0000 0.6000 -0.1533 0.0965 2.3021 0.0965 7.0000 0.7000 0.3262 0.6388 5.4229 0.6388 8.0000 0.8000 0.4445 0.9806 3.4175 0.9806 9.0000 0.9000 0.1676 0.8707 -1.0983 0.8707 10.0000 1.0000 -0.2001 0.4835 -3.8719 0.4835 11.0000 1.1000 -0.3329 0.1949 -2.8866 0.1949 12.0000 1.2000 -0.1616 0.2296 0.3476 0.2296 13.0000 1.3000 0.1145 0.5000 2.7033 0.5000 14.0000 1.4000 0.2443 0.7358 2.3586 0.7358 15.0000 1.5000 0.1451 0.7451 0.0929 0.7451 16.0000 1.6000 -0.0581 0.5611 -1.8402 0.5611 17.0000 1.7000 -0.1756 0.3735 -1.8754 0.3735 18.0000 1.8000 -0.1241 0.3408 -0.3270 0.3408 19.0000 1.9000 0.0224 0.4624 1.2156 0.4624 20.0000 2.0000 0.1236 0.6080 1.4564 0.6080 21.0000 2.1000 0.1025 0.6509 0.4287 0.6509 22.0000 2.2000 -0.0010 0.5736 -0.7732 0.5736 23.0000 2.3000 -0.0848 0.4629 -1.1070 0.4629 24.0000 2.4000 -0.0822 0.4180 -0.4494 0.4180 25.0000 2.5000 -0.0109 0.4647 0.4670 0.4647 26.0000 2.6000 0.0566 0.5471 0.8245 0.5471 27.0000 2.7000 0.0643 0.5895 0.4239 0.5895 28.0000 2.8000 0.0165 0.5634 -0.2606 0.5634 29.0000 2.9000 -0.0365 0.5032 -0.6018 0.5032 30.0000 3.0000 -0.0492 0.4657 -0.3751 0.4657 31.0000 3.1000 -0.0182 0.4783 0.1260 0.4783 32.0000 3.2000 0.0225 0.5214 0.4303 0.5214 33.0000 3.3000 0.0369 0.5531 0.3176 0.5531 34.0000 3.4000 0.0177 0.5489 -0.0418 0.5489 35.0000 3.5000 -0.0129 0.5189 -0.3009 0.5189 36.0000 3.6000 -0.0271 0.4929 -0.2600 0.4929 37.0000 3.7000 -0.0159 0.4921 -0.0079 0.4921 38.0000 3.8000 0.0066 0.5126 0.2051 0.5126 39.0000 3.9000 0.0195 0.5333 0.2070 0.5333 40.0000 4.0000 0.0136 0.5367 0.0345 0.5367 41.0000 4.1000 -0.0026 0.5232 -0.1358 0.5232 42.0000 4.2000 -0.0137 0.5071 -0.1610 0.5071 43.0000 4.3000 -0.0113 0.5024 -0.0462 0.5024 44.0000 4.4000 0.0002 0.5111 0.0866 0.5111 45.0000 4.5000 0.0095 0.5234 0.1225 0.5234 46.0000 4.6000 0.0091 0.5282 0.0489 0.5282 47.0000 4.7000 0.0011 0.5230 -0.0525 0.5230 48.0000 4.8000 -0.0063 0.5139 -0.0914 0.5139 49.0000 4.9000 -0.0071 0.5092 -0.0464 0.5092 50.0000 5.0000 -0.0018 0.5122 0.0295 0.5122 51.0000 5.1000 0.0041 0.5188 0.0668 0.5188 52.0000 5.2000 0.0054 0.5230 0.0412 0.5230 53.0000 5.3000 0.0020 0.5215 -0.0144 0.5215 54.0000 5.4000 -0.0025 0.5167 -0.0478 0.5167 55.0000 5.5000 -0.0041 0.5132 -0.0349 0.5132 56.0000 5.6000 -0.0019 0.5137 0.0050 0.5137 57.0000 5.7000 0.0015 0.5171 0.0335 0.5171 58.0000 5.8000 0.0030 0.5200 0.0286 0.5200 59.0000 5.9000 0.0017 0.5200 0.0006 0.5200 60.0000 6.0000 -0.0008 0.5177 -0.0229 0.5177 61.0000 6.1000 -0.0022 0.5154 -0.0229 0.5154 62.0000 6.2000 -0.0015 0.5151 -0.0036 0.5151


B =

0 0 0 0.1000 0.7614 0.7064 0.2000 0.7445 0.7052 0.3000 0.1045 0.1257 0.4000 -0.5073 -0.4400 0.5000 -0.5816 -0.5186 0.6000 -0.1533 -0.1323 0.7000 0.3262 0.3097 0.8000 0.4445 0.4256 0.9000 0.1676 0.1771 1.0000 -0.2001 -0.1608 1.1000 -0.3329 -0.2878 1.2000 -0.1616 -0.1352 1.3000 0.1145 0.1179 1.4000 0.2443 0.2405 1.5000 0.1451 0.1531 1.6000 -0.0581 -0.0327 1.7000 -0.1756 -0.1428 1.8000 -0.1241 -0.0984 1.9000 0.0224 0.0352 2.0000 0.1236 0.1295 2.1000 0.1025 0.1123 2.2000 -0.0010 0.0182 2.3000 -0.0848 -0.0596 2.4000 -0.0822 -0.0587 2.5000 -0.0109 0.0059 2.6000 0.0566 0.0683 2.7000 0.0643 0.0765 2.8000 0.0165 0.0333 2.9000 -0.0365 -0.0155 3.0000 -0.0492 -0.0280 3.1000 -0.0182 -0.0001 3.2000 0.0225 0.0372 3.3000 0.0369 0.0510 3.4000 0.0177 0.0339 3.5000 -0.0129 0.0059 3.6000 -0.0271 -0.0075 3.7000 -0.0159 0.0023 3.8000 0.0066 0.0229 3.9000 0.0195 0.0350 4.0000 0.0136 0.0300 4.1000 -0.0026 0.0151 4.2000 -0.0137 0.0047 4.3000 -0.0113 0.0068 4.4000 0.0002 0.0172 4.5000 0.0095 0.0258 4.6000 0.0091 0.0256 4.7000 0.0011 0.0184 4.8000 -0.0063 0.0115 4.9000 -0.0071 0.0107 5.0000 -0.0018 0.0155 5.1000 0.0041 0.0209 5.2000 0.0054 0.0222 5.3000 0.0020 0.0191 5.4000 -0.0025 0.0150 5.5000 -0.0041 0.0135 5.6000 -0.0019 0.0154 5.7000 0.0015 0.0185 5.8000 0.0030 0.0200 5.9000 0.0017 0.0189 6.0000 -0.0008 0.0166 6.1000 -0.0022 0.0153 6.2000 -0.0015 0.0158


B =

0 0 0 0.1000 0.7614 0.7892 0.2000 0.7445 0.7642 0.3000 0.1045 0.0937 0.4000 -0.5073 -0.5415 0.5000 -0.5816 -0.6136 0.6000 -0.1533 -0.1640 0.7000 0.3262 0.3346 0.8000 0.4445 0.4540 0.9000 0.1676 0.1626 1.0000 -0.2001 -0.2202 1.1000 -0.3329 -0.3558 1.2000 -0.1616 -0.1751 1.3000 0.1145 0.1127 1.4000 0.2443 0.2461 1.5000 0.1451 0.1409 1.6000 -0.0581 -0.0711 1.7000 -0.1756 -0.1924 1.8000 -0.1241 -0.1373 1.9000 0.0224 0.0158 2.0000 0.1236 0.1204 2.1000 0.1025 0.0974 2.2000 -0.0010 -0.0108 2.3000 -0.0848 -0.0977 2.4000 -0.0822 -0.0941 2.5000 -0.0109 -0.0194 2.6000 0.0566 0.0506 2.7000 0.0643 0.0580 2.8000 0.0165 0.0078 2.9000 -0.0365 -0.0473 3.0000 -0.0492 -0.0600 3.1000 -0.0182 -0.0274 3.2000 0.0225 0.0149 3.3000 0.0369 0.0296 3.4000 0.0177 0.0093 3.5000 -0.0129 -0.0226 3.6000 -0.0271 -0.0371 3.7000 -0.0159 -0.0253 3.8000 0.0066 -0.0017 3.9000 0.0195 0.0116 4.0000 0.0136 0.0053 4.1000 -0.0026 -0.0117 4.2000 -0.0137 -0.0232 4.3000 -0.0113 -0.0205 4.4000 0.0002 -0.0085 4.5000 0.0095 0.0011 4.6000 0.0091 0.0006 4.7000 0.0011 -0.0077 4.8000 -0.0063 -0.0155 4.9000 -0.0071 -0.0162 5.0000 -0.0018 -0.0106 5.1000 0.0041 -0.0045 5.2000 0.0054 -0.0032 5.3000 0.0020 -0.0068 5.4000 -0.0025 -0.0115 5.5000 -0.0041 -0.0131 5.6000 -0.0019 -0.0108 5.7000 0.0015 -0.0073 5.8000 0.0030 -0.0057 5.9000 0.0017 -0.0070 6.0000 -0.0008 -0.0096 6.1000 -0.0022 -0.0111 6.2000 -0.0015 -0.0104


B =

0 0 0 0.1000 0.7614 0.7437 0.2000 0.7445 0.7265 0.3000 0.1045 0.1007 0.4000 -0.5073 -0.4970 0.5000 -0.5816 -0.5691 0.6000 -0.1533 -0.1503 0.7000 0.3262 0.3183 0.8000 0.4445 0.4336 0.9000 0.1676 0.1626 1.0000 -0.2001 -0.1967 1.1000 -0.3329 -0.3262 1.2000 -0.1616 -0.1586 1.3000 0.1145 0.1113 1.4000 0.2443 0.2379 1.5000 0.1451 0.1408 1.6000 -0.0581 -0.0578 1.7000 -0.1756 -0.1725 1.8000 -0.1241 -0.1221 1.9000 0.0224 0.0212 2.0000 0.1236 0.1200 2.1000 0.1025 0.0992 2.2000 -0.0010 -0.0019 2.3000 -0.0848 -0.0838 2.4000 -0.0822 -0.0811 2.5000 -0.0109 -0.0114 2.6000 0.0566 0.0546 2.7000 0.0643 0.0620 2.8000 0.0165 0.0152 2.9000 -0.0365 -0.0366 3.0000 -0.0492 -0.0489 3.1000 -0.0182 -0.0186 3.2000 0.0225 0.0212 3.3000 0.0369 0.0352 3.4000 0.0177 0.0164 3.5000 -0.0129 -0.0135 3.6000 -0.0271 -0.0273 3.7000 -0.0159 -0.0164 3.8000 0.0066 0.0057 3.9000 0.0195 0.0182 4.0000 0.0136 0.0125 4.1000 -0.0026 -0.0034 4.2000 -0.0137 -0.0143 4.3000 -0.0113 -0.0119 4.4000 0.0002 -0.0006 4.5000 0.0095 0.0084 4.6000 0.0091 0.0080 4.7000 0.0011 0.0003 4.8000 -0.0063 -0.0070 4.9000 -0.0071 -0.0078 5.0000 -0.0018 -0.0026 5.1000 0.0041 0.0032 5.2000 0.0054 0.0045 5.3000 0.0020 0.0011 5.4000 -0.0025 -0.0033 5.5000 -0.0041 -0.0048 5.6000 -0.0019 -0.0027 5.7000 0.0015 0.0006 5.8000 0.0030 0.0021 5.9000 0.0017 0.0009 6.0000 -0.0008 -0.0016 6.1000 -0.0022 -0.0030 6.2000 -0.0015 -0.0023


B =

0 0 0 0.1000 0.7614 0.8243 0.2000 0.7445 0.8723 0.3000 0.1045 0.2443 0.4000 -0.5073 -0.4090 0.5000 -0.5816 -0.5336 0.6000 -0.1533 -0.1206 0.7000 0.3262 0.3853 0.8000 0.4445 0.5418 0.9000 0.1676 0.2806 1.0000 -0.2001 -0.1029 1.1000 -0.3329 -0.2640 1.2000 -0.1616 -0.1076 1.3000 0.1145 0.1773 1.4000 0.2443 0.3277 1.5000 0.1451 0.2416 1.6000 -0.0581 0.0342 1.7000 -0.1756 -0.0980 1.8000 -0.1241 -0.0576 1.9000 0.0224 0.0903 2.0000 0.1236 0.2017 2.1000 0.1025 0.1896 2.2000 -0.0010 0.0866 2.3000 -0.0848 -0.0043 2.4000 -0.0822 -0.0088 2.5000 -0.0109 0.0612 2.6000 0.0566 0.1334 2.7000 0.0643 0.1466 2.8000 0.0165 0.1004 2.9000 -0.0365 0.0445 3.0000 -0.0492 0.0276 3.1000 -0.0182 0.0569 3.2000 0.0225 0.0993 3.3000 0.0369 0.1169 3.4000 0.0177 0.0993 3.5000 -0.0129 0.0677 3.6000 -0.0271 0.0512 3.7000 -0.0159 0.0610 3.8000 0.0066 0.0840 3.9000 0.0195 0.0985 4.0000 0.0136 0.0939 4.1000 -0.0026 0.0774 4.2000 -0.0137 0.0652 4.3000 -0.0113 0.0666 4.4000 0.0002 0.0781 4.5000 0.0095 0.0881 4.6000 0.0091 0.0885 4.7000 0.0011 0.0807 4.8000 -0.0063 0.0727 4.9000 -0.0071 0.0714 5.0000 -0.0018 0.0765 5.1000 0.0041 0.0827 5.2000 0.0054 0.0845 5.3000 0.0020 0.0812 5.4000 -0.0025 0.0765 5.5000 -0.0041 0.0746 5.6000 -0.0019 0.0766 5.7000 0.0015 0.0801 5.8000 0.0030 0.0819 5.9000 0.0017 0.0808 6.0000 -0.0008 0.0783 6.1000 -0.0022 0.0767 6.2000 -0.0015 0.0772


B =

0 1.0000 0 0.1000 1.7614 0.8243 0.2000 1.7445 0.8723 0.3000 1.1045 0.2443 0.4000 0.4927 -0.4090 0.5000 0.4184 -0.5336 0.6000 0.8467 -0.1206 0.7000 1.3262 0.3853 0.8000 1.4445 0.5418 0.9000 1.1676 0.2806 1.0000 0.7999 -0.1029 1.1000 0.6671 -0.2640 1.2000 0.8384 -0.1076 1.3000 1.1145 0.1773 1.4000 1.2443 0.3277 1.5000 1.1451 0.2416 1.6000 0.9419 0.0342 1.7000 0.8244 -0.0980 1.8000 0.8759 -0.0576 1.9000 1.0224 0.0903 2.0000 1.1236 0.2017 2.1000 1.1025 0.1896 2.2000 0.9990 0.0866 2.3000 0.9152 -0.0043 2.4000 0.9178 -0.0088 2.5000 0.9891 0.0612 2.6000 1.0566 0.1334 2.7000 1.0643 0.1466 2.8000 1.0165 0.1004 2.9000 0.9635 0.0445 3.0000 0.9508 0.0276 3.1000 0.9818 0.0569 3.2000 1.0225 0.0993 3.3000 1.0369 0.1169 3.4000 1.0177 0.0993 3.5000 0.9871 0.0677 3.6000 0.9729 0.0512 3.7000 0.9841 0.0610 3.8000 1.0066 0.0840 3.9000 1.0195 0.0985 4.0000 1.0136 0.0939 4.1000 0.9974 0.0774 4.2000 0.9863 0.0652 4.3000 0.9887 0.0666 4.4000 1.0002 0.0781 4.5000 1.0095 0.0881 4.6000 1.0091 0.0885 4.7000 1.0011 0.0807 4.8000 0.9937 0.0727 4.9000 0.9929 0.0714 5.0000 0.9982 0.0765 5.1000 1.0041 0.0827 5.2000 1.0054 0.0845 5.3000 1.0020 0.0812 5.4000 0.9975 0.0765 5.5000 0.9959 0.0746 5.6000 0.9981 0.0766 5.7000 1.0015 0.0801 5.8000 1.0030 0.0819 5.9000 1.0017 0.0808 6.0000 0.9992 0.0783 6.1000 0.9978 0.0767 6.2000 0.9985 0.0772

Cdigo de las funciones de los mtodos usados

Cdigo para el Mtodo de Punto Medio:

function Mid_Method(f,ti,tf,Xi,Yi,h)n=(tf-ti)/h;x1=zeros(n,1);y1=zeros(n,1);x1=[ti:h:tf];y1(1)=Yi;y2(1)=Yi;A(1)=0;for k=1:n x=x1(k); y=y1(k); fp(k+1)=eval(f); y2(k+1)=y1(k)+((fp(k+1))*h)/2; y=y2(k+1); x=x1(k)+h/2; fp2(k+1)=eval(f); y1(k+1)=y1(k)+(fp2(k+1))*h; A(k+1,1)=k;endplot(x1,y1,'-m')primitiva=int(sym(f));Q=length(x1);for u=1:Q x=x1(u); y4(u)=eval(primitiva);enddisp(' x Y_Verdadero Y_Punto_Medio')B=[x1', y4', y1]end

Cdigo para el Mtodo de Runge Kutta de 2do. Orden (Ralston):

function Runge_Kutta2_Ralston(f,ti,tf,Xi,Yi,h)n=(tf-ti)/h;x1=zeros(n,1);y1=zeros(n,1);x1=[ti:h:tf];y1(1)=Yi;y2(1)=Yi;A(1)=0;for k=1:n x=x1(k); y=y1(k); k1(k+1)=eval(f); y2(k+1)=y1(k)+((k1(k+1))*3*h)/4; y=y2(k+1); x=x1(k)+3*h/4; k2(k+1)=eval(f); y1(k+1)=y1(k)+((1/3)*k1(k+1)+(2/3)*(k2(k+1)))*h; A(k+1,1)=k;endplot(x1,y1,'-c')primitiva=int(sym(f));Q=length(x1);for u=1:Q x=x1(u); y4(u)=eval(primitiva);enddisp(' x Y_Verdadero Y_Runge_Kutta2_Ralston')B=[x1', y4', y1]end

Cdigo para el Mtodo de Runge Kutta de 3er. Orden:

function Runge_Kutta3(f,ti,tf,Xi,Yi,h)n=(tf-ti)/h;x1=zeros(n,1);y1=zeros(n,1);x1=[ti:h:tf]y1(1)=Yi;y2(1)=Yi;A(1)=0;for k=1:n x=x1(k); y=y1(k); k1(k+1)=eval(f); y2(k+1)=y1(k)+((k1(k+1))*h)/2; y=y2(k+1); x=x1(k)+h/2; k2(k+1)=eval(f); y3(k+1)=y1(k)-h*k1(k+1)+2*h*k2(k+1); y=y3(k+1); x=x1(k)+h/2; k3(k+1)=eval(f); y1(k+1)=y1(k)+(1/6)*(k1(k+1)+4*(k2(k+1))+(k3(k+1)))*h; A(k+1,1)=k;endplot(x1,y1,'--k')primitiva=int(sym(f));Q=length(x1);for u=1:Q x=x1(u); y4(u)=eval(primitiva);enddisp(' x Y_Verdadero Y_Runge_Kutta3')B=[x1', y4', y1]end

Cdigo para el Mtodo de Runge Kutta de 4to. Orden:

function Runge_Kutta4(f,ti,tf,Xi,Yi,h)n=(tf-ti)/h;x1=zeros(n,1);y1=zeros(n,1);x1=[ti:h:tf]y1(1)=Yi;y2(1)=Yi;A(1)=0;for k=1:n x=x1(k); y=y1(k); k1(k+1)=eval(f); y2(k+1)=y1(k)+((k1(k+1))*h)/2; y=y2(k+1); x=x1(k)+h/2; k2(k+1)=eval(f); y3(k+1)=y1(k)+((k2(k+1))*h)/2; y=y3(k+1); x=x1(k)+h/2; k3(k+1)=eval(f); y4(k+1)=y1(k)+(k3(k+1))*h; y=y4(k+1); x=x1(k)+h/2; k4(k+1)=eval(f); y1(k+1)=y1(k)+(1/6)*(k1(k+1)+2*(k2(k+1))+2*(k3(k+1))+k4(k+1))*h; A(k+1,1)=k;endplot(x1,y1,'-y')primitiva=int(sym(f))+1Q=length(x1);for u=1:Q x=x1(u); y4(u)=eval(primitiva);enddisp(' x Y_Verdadero Y_Runge_Kutta4')B=[x1', y4', y1]end

Curso Propedutico1

Tarea II. Circuitos Eléctricos. Propedeútico UMSNH

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Transcript of Tarea II. Circuitos Eléctricos. Propedeútico UMSNH