{"id":33463,"date":"2022-03-15T13:00:18","date_gmt":"2022-03-15T13:00:18","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33463"},"modified":"2025-07-26T06:21:06","modified_gmt":"2025-07-26T06:21:06","slug":"introductio-ad-aequationes-differentiales-ordinarias","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/introductio-ad-aequationes-differentiales-ordinarias\/","title":{"rendered":"Introductio ad Aequationes Differentiales Ordinarias"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>Introductio ad Aequationes Differentiales Ordinarias<\/h1>\n<p style=\"text-align:center;\"><em>In hac classe exploratio accurata idearum fundamentalium quae has aequationes regunt earumque applicationes in variis campis offertur. Incipiendo ab analysi naturae mutationis incessantis in mundo qui nos circumdat, exhibentur notiones fundamentales sicut functiones, derivativae earumque relatio cum mutatione continua et discreta. Introducta fit distinctio inter Aequationes Differentiales Partiales (EDP) et Ordinarias (EDO), studio EDO potissimum spectato. Notiones illustrantur exemplis practicis ut refrigeratio calicis caffeae, Leges Newtonianae et exempla populationis. Studiosis dabitur opportunitas cognoscendi aequationes differentiales quae phaenomena naturalia et physica regunt, quo modo mathematice repraesentari possint detegendi et quasdam technicas ad studia solutionum earum intellegendi. Haec cognitio initialis fundamentum constituet ad studia provectiora in aequationibus differentialibus earumque applicationibus in scientia et machinatione..<\/em><\/p>\n<p style=\"text-align:center;\"><strong><u>Proposita Discendi<\/u>:<\/strong><br \/>Hoc cursu confecto discipulus poterit:<\/p>\n<ol>\n<li><strong>Intelligere<\/strong> notiones fundamentales quae cum aequationibus differentialibus coniunguntur, ut naturam mutationis, functiones, derivativas et differentias inter Aequationes Differentiales Partiales (EDP) et Ordinarias (EDO)<\/li>\n<\/ul>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/bYwm6NAEvVA\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><br \/>\n<center><\/p>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#LasEcuacionesDiferencialesYLaNaturalezaDeLasCosas\"><strong>Aequationes Differentiales et Natura Rerum<\/strong><\/a><br \/>\n<a href=\"#ElCambioIncesante\">Mutatio Incessans<\/a><br \/>\n<a href=\"#FuncionesDerivadasYSusCambios\">Functiones, derivatae et earum mutationes<\/a><br \/>\n<a href=\"#EDOyEDP\">EDO et EDP<\/a><br \/>\n<a href=\"#EjemplosDeEcuacionesDiferencialesOrdinarias\"><strong>Exempla Aequationum Differentialium Ordinariarum<\/strong><\/a><br \/>\n<a href=\"#ElEnfriamientoDeUnaTazaDeCafe\">Refrigeratio poculi coffeae<\/a><br \/>\n<a href=\"#LasLeyesDeNewton\">Leges Newtonianae<\/a><br \/>\n<a href=\"#ModeloDePoblaciones\">Exemplar Populationum<\/a>\n<\/p>\n<p><\/center><\/p>\n<p><a name=\"LasEcuacionesDiferencialesYLaNaturalezaDeLasCosas\"><\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/KgUDA2Q1qaA\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<h2>Aequationes Differentiales et Natura Rerum<\/h2>\n<p><a name=\"ElCambioIncesante\"><\/a><\/p>\n<h3>Mutatio Incessans<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=133s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">In natura omnia in perpetua mutatione sunt.<\/span><\/strong><\/a> Etiam quod invariabile videri potest, ut fulgor Solis, variat si in apta temporis scala observetur. Omnia mutantur: fulgor stellarum, temperatura coffeae in poculo, positio obiecti et magnitudo populationis sunt nonnulla exempla; et hae mutationis rationes plerumque cum statu illius quod mutatur dum mutatio fit coniunguntur.<\/p>\n<p>Una via intuitive ad mutationem intellegendam est observare quomodo res se modificent dum tempus transit. Mutationem quae ad tempus refertur appellamus evolutionem, et omnia quae observamus in continua evolutione sunt. Sed evolutio sola forma mutationis non est; exempli causa, licet altitudo nostra a gradu maris tempore mutari possit, verius est eam secundum situm nostrum (vel coordinatas geographicas) mutari.<\/p>\n<p><a name=\"FuncionesDerivadasYSusCambios\"><\/a><\/p>\n<h3>Functiones, derivatae et earum mutationes<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=301s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Generaliter,<\/span><\/strong><\/a> functio plurium variabilium <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_1,x_2, \\cdots, x_n)<\/span><\/span> variari potest si aliqua earum variabilium mutetur, et ista mutatio continua vel discreta esse potest. Pro functione plurium variabilium, mutatio continua per <strong>derivatas partiales:<\/strong> examinari potest:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial f(x_1, \\cdots, x_n)}{\\partial x_1} = \\lim_{\\Delta x_1 \\to 0} \\frac{ f(x_1 + \\Delta x_1, \\cdots, x_n) -  f(x_1, \\cdots, x_n)}{\\Delta x_1} <\/span>\n<p>Si functio unius variabilis est, adhibetur <strong>derivata ordinaria:<\/strong><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} = \\lim_{\\Delta x \\to 0} \\frac{ f(x + \\Delta x) -  f(x)}{\\Delta x} <\/span>\n<p>Si mutatio discreta est potius quam continua, simpliciter omittitur calculus limi qui in derivatis apparet.<\/p>\n<p><a name=\"EDOyEDP\"><\/a><\/p>\n<h3>EDO et EDP<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=624s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Aequatio quae functionem eiusque diversas derivatas involvit<\/span><\/strong><\/a> vocatur <strong>Aequatio Differentialis<\/strong>. Si hae derivatae partiales sive ordinariae sunt, respective <strong>Aequationes Differentiales Partiales (EDP)<\/strong> vel <strong>Aequationes Differentiales Ordinariae (EDO)<\/strong> appellantur. Hoc tempore, in studio aequationum differentialium ordinariarum intendemus et quaedam exempla in quibus occurrunt recognoscemus.<\/p>\n<p><a name=\"EjemplosDeEcuacionesDiferencialesOrdinarias\"><\/a><\/p>\n<h2>Exempla Aequationum Differentialium Ordinariarum<\/h2>\n<p><a name=\"ElEnfriamientoDeUnaTazaDeCafe\"><\/a><\/p>\n<h3>Refrigeratio poculi coffeae<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=680s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Ratio refrigerandi poculum coffeae proportionalis est<\/span><\/strong><\/a> differentiae temperaturae inter ambitum et coffeam. Si temperatura aeris, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_a<\/span><\/span>, constans est et temperatura coffeae est functio temporis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_c=T_c(t),<\/span><\/span>, aequationem differentialem invenire possumus quae nobis sinet temperaturam coffeae in quolibet momento determinare. Initio habemus:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dT_c(t)}{dt} = -\\alpha^2(T_c(t) - T_a) <\/span>\n<p>Ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> est constans proportionalitatis, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_a \\lt T_c(t)<\/span><\/span> et signum negativum indicat temperaturam coffeae decrescere. Postea videbimus hanc aequationem solutionem habere huius formae:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_c(t) = T_a + Be^{-\\alpha^2 t}<\/span>\n<p>Ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> est constans determinanda.<\/p>\n<p><a name=\"LasLeyesDeNewton\"><\/a><\/p>\n<h3>Leges Newtonianae<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=885s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Secunda Lex Newtoniana essentialiter est aequatio differentialis ordinaria,<\/span><\/strong><\/a> quia in enuntiatione <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F=ma<\/span><\/span> (vis aequalis est massae multiplicatae per accelerationem), acceleratio, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=d^2x(t)\/dt^2,<\/span><\/span>, est secunda derivata temporalis positionis obiecti. Per hanc legem relationes invenire possumus quae motum corporum describunt, quae re vera aequationes differentiales sunt. Exemplum simplex est studium ressortium: si funiculus elasticus uno latere muro fixo altero massae in statu aequilibrii affixus est et massam deinde ab hoc situ per spatium <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> removemus, lege Hooke massa vim restitutionis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F=-kx<\/span><\/span> sentiet. Tum, per secundam legem Newtonianam, habebimus:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle -kx(t) = m\\frac{d^2x(t)}{dt^2} <\/span>\n<p>Postea constabimus solutionem eius talis formae esse:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x(t) = A\\sin\\left(\\sqrt{\\frac{k}{m}}t + \\phi \\right)<\/span>\n<p>Ubi <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> constantiae sunt quae ab <strong>condicionibus initialibus problematis<\/strong> determinabuntur.<\/p>\n<p><a name=\"ModeloDePoblaciones\"><\/a><\/p>\n<h3>Exemplar Populationum<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=1184s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Ratio incrementi per habitantem<\/span><\/strong><\/a> populationis est aequalis differentiae inter ratem natalitatis et mortis, id est:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{1}{x(t)} \\frac{dx(t)}{dt} = N - M<\/span>\n<p>Si rata natalium <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> tempore constans manet et mortes proportionale sunt populationi, id est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M=\\alpha^2 x(t),<\/span><\/span>, tunc aequatio praecedens talem formam accipit:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx(t)}{dt} = x(t) (N - \\alpha^2 x(t))<\/span>\n<p>Hoc notum est ut <strong>\u00abAequatio Logistica Populationum\u00bb<\/strong>. Ex hac aequatione generalizatio construi potest pro multis populationibus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1(t), x_2(t), \\cdots, x_n(t)<\/span><\/span> quae inter se de exsistentia certant hoc modo:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx_i(t)}{dt} = x_i(t) \\left(N_i - \\displaystyle \\sum_{j=1}^n\\alpha^2_{ij} x_j(t)  \\right)<\/span>\n<p>Cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\in\\{1,\\cdots, n\\}<\/span><\/span>. Hoc est quod dicitur <strong>Aequationes Lotka-Volterrae<\/strong>.<\/p>\n<h2>Conclusio<\/h2>\n<p>Per hanc introductionem ad Aequationes Differentiales Ordinarias exploravimus quomodo mathematicae mutationes quae in mundo naturali occurrunt accurate et eleganter capere possint. A refrigeratione poculi coffeae usque ad motum ressortis vel incrementum populationis, EDO permittunt dynamicas complexas in relationes mathematice intellegibiles et examinabiles convertere.<\/p>\n<p>Intellegere structuram et significationem harum aequationum ianuam aperit multis disciplinis, sicut physicae, biologiae, oeconomiae et machinationis. Haec classis fundamenta conceptuum necessaria ponit ad studia provectiora continuanda, ubi technicae solutionum, analysis qualitativus et methodi numerici altius pertractabuntur. Maxime tamen momenti est primam quandam intuitionem evolvere de quo modo lingua mutationis \u2014aequationes differentiales\u2014 nos sinat describere, intellegere et praedicere comportamentum systematum dynamicorum.<\/p>\n<p>In subsequentibus disciplinis instrumenta potentiora evolvemus eaque novis contextibus applicabimus. Aequationes differentiales non solum modum praebent ad realitatem analysi subiciendam, sed etiam ad imaginandum quomodo ea sub diversis condicionibus evolvere possit.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Introductio ad Aequationes Differentiales Ordinarias In hac classe exploratio accurata idearum fundamentalium quae has aequationes regunt earumque applicationes in variis campis offertur. Incipiendo ab analysi naturae mutationis incessantis in mundo qui nos circumdat, exhibentur notiones fundamentales sicut functiones, derivativae earumque relatio cum mutatione continua et discreta. Introducta fit distinctio inter Aequationes Differentiales Partiales (EDP) et [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32792,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":4,"footnotes":""},"categories":[1346,1298],"tags":[],"class_list":["post-33463","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-aequationes-differentiae-ordinariae","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Introductio ad Aequationes Differentiales Ordinarias - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explora quomodo aequationes differentiales ordinariae phaenomena realia sicut motus, calor vel incrementum populationis explicent.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/la\/introductio-ad-aequationes-differentiales-ordinarias\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Introductio ad Aequationes Differentiales Ordinarias\" \/>\n<meta property=\"og:description\" content=\"Explora quomodo aequationes differentiales ordinariae phaenomena realia sicut motus, calor vel incrementum populationis explicent.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/introductio-ad-aequationes-differentiales-ordinarias\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2022-03-15T13:00:18+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-26T06:21:06+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/edointroflat-1024x381.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Introductio ad Aequationes Differentiales Ordinarias\" \/>\n<meta name=\"twitter:description\" content=\"Explora quomodo aequationes differentiales ordinariae phaenomena realia sicut motus, calor vel incrementum populationis explicent.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/edointroflat.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/introductio-ad-aequationes-differentiales-ordinarias\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/introductio-ad-aequationes-differentiales-ordinarias\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Introductio ad Aequationes Differentiales Ordinarias\",\"datePublished\":\"2022-03-15T13:00:18+00:00\",\"dateModified\":\"2025-07-26T06:21:06+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/introductio-ad-aequationes-differentiales-ordinarias\\\/\"},\"wordCount\":1010,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/introductio-ad-aequationes-differentiales-ordinarias\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/edointroflat.jpg\",\"articleSection\":[\"Aequationes Differentiae Ordinariae\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/introductio-ad-aequationes-differentiales-ordinarias\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/introductio-ad-aequationes-differentiales-ordinarias\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/introductio-ad-aequationes-differentiales-ordinarias\\\/\",\"name\":\"Introductio ad Aequationes Differentiales Ordinarias - 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