{"id":1042,"date":"2018-02-09T10:43:50","date_gmt":"2018-02-09T15:43:50","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/matg1013\/?p=1042"},"modified":"2026-04-05T20:15:39","modified_gmt":"2026-04-06T01:15:39","slug":"s2eva2017tii_t1-edo-runge-kutta-2do-orden-d2y-dx2","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/mn-s2eva20\/s2eva2017tii_t1-edo-runge-kutta-2do-orden-d2y-dx2\/","title":{"rendered":"s2Eva2017TII_T1 EDO Runge Kutta 2do Orden d2y\/dx2"},"content":{"rendered":"\n<p><strong>Ejercicio<\/strong>: <a href=\"https:\/\/blog.espol.edu.ec\/algoritmos101\/mn-2eva20\/2eva2017tii_t1-edo-runge-kutta-2do-orden-d2ydx2\/\" data-type=\"post\" data-id=\"1029\">2Eva2017TII_T1 EDO Runge Kutta 2do Orden d2y\/dx2<\/a><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Tema 1<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>Runge kutta de 2do Orden\nf: y' = z\ng: z' = .....\nK1y = h f(xi, yi, zi)\nK1z = h g(xi, y1, zi)\n\nK2y = h f(xi+h, yi+K1y, zi+K1z)\nK2z = h g(xi+h, yi+K1y, zi+K1z)\n\ny(i+1) = yi + (1\/2)(K1y + K2y)\nz(i+1) = zi + (1\/2)(K1z + K2z)\n\nx(i+1) = xi + h\nE = O(h<sup>3<\/sup>) \nxi \u2264 z \u2264 x(i+1)<\/code><\/pre>\n\n\n\n<p>f: z = \u0398'<br>g: z' = (-gr\/L) sin(\u0398)<\/p>\n\n\n\n<p>\u0398(0) = \u03c0\/6<br>z(0) = 0<\/p>\n\n\n\n<p>h=0.1<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>i=0, t0 = 0, \u03980 = \u03c0\/6, z0 = 0\n    K1y = 0.1(0) = 0\n    K1z = 0.1(-9.8\/2)sin(\u03c0\/6) = -0.245\n\n    K2y = 0.1(0+(-0.245)) = -0.0245\n    K2z = 0.1(-9.8\/2)sin(\u03c0\/6+0) = -0.245\n\n    \u03981 = \u03c0\/6 + (1\/2)(0+(-0.0245)) = 0.51139\n    z1 = 0 + (1\/2)(-0.245-0.245) = -0.245\n    t1 = 0 + 0.1 = 0.1\n\ni=1, t1 = 0.1, \u03981 = 0.51139, z1 = -0.245\n    K1y = 0.1(-0.245) = -0.0245\n    K1z = 0.1(-9.8\/2)sin(0.51139) = -0.23978\n\n    K2y = 0.1(-0.245+(-0.0245)) = -0.049\n    K2z = 0.1(-9.8\/2)sin(0.51139+(-0.0245)) = -0.22924\n\n    \u03982 = 0.51139 + (1\/2)(-0.0245+(-0.049)) = 0.47509\n    z2 = -0.245 + (1\/2)(-0.23978+(-0.22924)) = -0.245\n    t2 = 0.1 + 0.1 = 0.2\n<\/code><\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"480\" src=\"http:\/\/blog.espol.edu.ec\/algoritmos101\/files\/2018\/02\/s2Eva2017TII_T1_RK2o_d2yd2x.png\" alt=\"s2Eva2017TII_T1Runge-Kutta 2do orden d2y\/dx2\" class=\"wp-image-18484\" \/><\/figure>\n\n\n\n<pre class=\"wp-block-code\"><code>   t         theta     z\n&#091;&#091; 0.        0.523599  0.      ]\n &#091; 0.1       0.511349 -0.245   ]\n &#091; 0.2       0.47486  -0.479513]\n &#091; 0.3       0.415707 -0.692975]\n &#091; 0.4       0.336515 -0.875098]\n &#091; 0.5       0.240915 -1.016375]\n &#091; 0.6       0.133432 -1.108842]\n &#091; 0.7       0.019289 -1.14696 ]\n &#091; 0.8      -0.09588  -1.128346]\n &#091; 0.9      -0.206369 -1.054127]\n &#091; 1.       -0.306761 -0.92877 ]\n &#091; 1.1      -0.39224  -0.759461]\n &#091; 1.2      -0.458821 -0.555246]\n &#091; 1.3      -0.503495 -0.326207]\n &#091; 1.4      -0.524294 -0.082851]\n &#091; 1.5      -0.520315  0.164197]\n &#091; 1.6      -0.491715  0.404296]\n &#091; 1.7      -0.439718  0.62682 ]\n &#091; 1.8      -0.366606  0.821313]\n &#091; 1.9      -0.275693  0.977893]\n &#091; 2.       -0.171235  1.087942]]<\/code><\/pre>\n\n\n\n<p>Literal b), con h= 0.25, con t = 1 \u00e1ngulo= -0.352484<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>   t         theta     z\n&#091;&#091; 0.        0.523599  0.      ]\n &#091; 0.25      0.447036 -0.6125  ]\n &#091; 0.5       0.227716 -1.054721]\n &#091; 0.75     -0.070533 -1.170971]\n &#091; 1.       -0.352484 -0.910162]\n &#091; 1.25     -0.527161 -0.363031]\n &#091; 1.5      -0.540884  0.299952]\n &#091; 1.75     -0.387053  0.890475]\n &#091; 2.       -0.106636  1.221932]]<\/code><\/pre>\n\n\n\n<p>El error de del orden h<sup>3<\/sup><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<h2 class=\"wp-block-heading\">Instrucciones en Python<\/h2>\n\n\n\n<p>usando el algoritmo desarrollado en clase<\/p>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: python; title: ; notranslate\" title=\"\">\n# Runge Kutta de 2do\n# EDO de 2do orden con condiciones de inicio\nimport numpy as np\nimport matplotlib.pyplot as plt\n\ndef rungekutta2_fg(f,g,v0,h,m):\n    tabla = &#x5B;v0]\n    xi = v0&#x5B;0]\n    yi = v0&#x5B;1]\n    zi = v0&#x5B;2]\n    for i in range(0,m,1):\n        K1y = h * f(xi,yi,zi)\n        K1z = h * g(xi,yi,zi)\n        \n        K2y = h * f(xi+h, yi + K1y, zi+K1z)\n        K2z = h * g(xi+h, yi + K1y, zi+K1z)\n\n        yi1 = yi + (1\/2)*(K1y+K2y)\n        zi1 = zi + (1\/2)*(K1z+K2z)\n        xi1 = xi + h\n        vector = &#x5B;xi1,yi1,zi1]\n        tabla.append(vector)\n\n        xi = xi1\n        yi = yi1\n        zi = zi1\n    tabla = np.array(tabla)\n    return(tabla)\n\n# Programa Prueba\n# Funciones\nf = lambda x,y,z : z\ng = lambda x,y,z : (-gr\/L)*np.sin(y)\n\ngr = 9.8\nL = 2\n\nx0 = 0\ny0 = np.pi\/6\nz0 = 0\n\nv0 = &#x5B;x0,y0,z0]\n\nh = 0.1\nxn = 2\nm = int((xn-x0)\/h)\n\n# PROCEDIMIENTO\ntabla = rungekutta2_fg(f,g,v0,h,m)\n\nxi = tabla&#x5B;:,0]\nyi = tabla&#x5B;:,1]\nzi = tabla&#x5B;:,2]\n\n# SALIDA\nnp.set_printoptions(precision=6)\nprint('x, y, z')\nprint(tabla)\nplt.plot(xi,yi, label='y')\nplt.plot(xi,zi, label='z')\nplt.legend()\nplt.grid()\nplt.show()\n<\/pre><\/div>","protected":false},"excerpt":{"rendered":"<p>Ejercicio: 2Eva2017TII_T1 EDO Runge Kutta 2do Orden d2y\/dx2 Tema 1 f: z = \u0398'g: z' = (-gr\/L) sin(\u0398) \u0398(0) = \u03c0\/6z(0) = 0 h=0.1 Literal b), con h= 0.25, con t = 1 \u00e1ngulo= -0.352484 El error de del orden h3 Instrucciones en Python usando el algoritmo desarrollado en clase<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"wp-custom-template-entrada-mn-ejemplo","format":"standard","meta":{"footnotes":""},"categories":[48],"tags":[58,54],"class_list":["post-1042","post","type-post","status-publish","format-standard","hentry","category-mn-s2eva20","tag-ejemplos-python","tag-mnumericos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1042","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/users\/8043"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/comments?post=1042"}],"version-history":[{"count":3,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1042\/revisions"}],"predecessor-version":[{"id":23867,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1042\/revisions\/23867"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=1042"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=1042"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=1042"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}