{"id":32794,"date":"2022-03-15T13:00:36","date_gmt":"2022-03-15T13:00:36","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32794"},"modified":"2025-04-01T18:27:43","modified_gmt":"2025-04-01T18:27:43","slug":"introduction-to-ordinary-differential-equations","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/introduction-to-ordinary-differential-equations\/","title":{"rendered":"Introduction to Ordinary Differential Equations"},"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>Introduction to Ordinary Differential Equations<\/h1>\n<p style=\"text-align:center;\"><em>This class offers a detailed exploration of the fundamental ideas that govern these equations and their applications in various fields. Beginning with an analysis of the nature of constant change in the world around us, basic concepts such as functions, derivatives, and their relationship with continuous and discrete change are presented. The distinction between Partial Differential Equations (PDEs) and Ordinary Differential Equations (ODEs) is introduced, focusing on the study of ODEs. Concepts are illustrated with practical examples such as the cooling of a cup of coffee, Newton&#8217;s Laws, and population models. Students will have the opportunity to become familiar with differential equations that govern natural and physical phenomena, discover how they can be mathematically represented, and understand some techniques to study their solutions. This initial knowledge will constitute the foundation for more advanced studies in differential equations and their applications in science and engineering.<\/em><\/p>\n<p style=\"text-align:center;\"><strong><u>Learning Objectives<\/u>:<\/strong><br \/>By the end of this class, the student will be able to:<\/p>\n<ol>\n<li><strong>Understand<\/strong> the basic concepts related to differential equations, such as the nature of change, functions, derivatives, and the differences between Partial Differential Equations (PDEs) and Ordinary Differential Equations (ODEs)<\/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>Differential Equations and the Nature of Things<\/strong><\/a><br \/>\n<a href=\"#ElCambioIncesante\">The Constant Change<\/a><br \/>\n<a href=\"#FuncionesDerivadasYSusCambios\">Functions, Derivatives, and Their Changes<\/a><br \/>\n<a href=\"#EDOyEDP\">ODEs and PDEs<\/a><br \/>\n<a href=\"#EjemplosDeEcuacionesDiferencialesOrdinarias\"><strong>Examples of Ordinary Differential Equations<\/strong><\/a><br \/>\n<a href=\"#ElEnfriamientoDeUnaTazaDeCafe\">The Cooling of a Cup of Coffee<\/a><br \/>\n<a href=\"#LasLeyesDeNewton\">Newton&#8217;s Laws<\/a><br \/>\n<a href=\"#ModeloDePoblaciones\">Population Model<\/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>Differential Equations and the Nature of Things<\/h2>\n<p><a name=\"ElCambioIncesante\"><\/a><\/p>\n<h3>The Constant Change<\/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 nature, everything is in constant change.<\/span><\/strong><\/a> Even things that seem to never change, like the brightness of the Sun, vary if observed on the appropriate time scale. Everything changes: the brightness of stars, the temperature of coffee in a cup, the position of an object, and the size of a population are some examples, and these rates of change are generally related to the state of what is changing while that change occurs.<\/p>\n<p>An intuitive way to understand change is to observe how things change over time. The change that occurs with respect to time is what we call evolution, and everything we can observe is in continuous evolution. But evolution is not the only form of change; for example, although our height above sea level may vary over time, it is more likely to change depending on our position (or geographic coordinates).<\/p>\n<p><a name=\"FuncionesDerivadasYSusCambios\"><\/a><\/p>\n<h3>Functions, Derivatives, and Their Changes<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=301s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">In more general terms,<\/span><\/strong><\/a> a function of several variables <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_1,x_2, \\cdots, x_n)<\/span><\/span> can vary if any of its variables change, and that change can be continuous or discrete. For a function of several variables, continuous change can be studied through <strong>partial derivatives:<\/strong><\/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>If the function is of a single variable, the <strong>ordinary derivative<\/strong> is used:<\/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>If the change is discrete instead of continuous, the calculation of the limit that appears in the derivatives is simply omitted.<\/p>\n<p><a name=\"EDOyEDP\"><\/a><\/p>\n<h3>ODEs and PDEs<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=624s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">An equation involving a function and its different derivatives<\/span><\/strong><\/a> is known as a <strong>Differential Equation<\/strong>. If these derivatives are partial or ordinary, they are called respectively <strong>Partial Differential Equations (PDEs)<\/strong> or <strong>Ordinary Differential Equations (ODEs)<\/strong>. At this point, we will focus on the study of ordinary differential equations and review some examples where they appear.<\/p>\n<p><a name=\"EjemplosDeEcuacionesDiferencialesOrdinarias\"><\/a><\/p>\n<h2>Examples of Ordinary Differential Equations<\/h2>\n<p><a name=\"ElEnfriamientoDeUnaTazaDeCafe\"><\/a><\/p>\n<h3>The Cooling of a Cup of Coffee<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=680s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">The cooling rate of a cup of coffee is proportional<\/span><\/strong><\/a> to the temperature difference between the environment and the coffee. If the air temperature, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_a<\/span><\/span>, is constant and the temperature of the coffee is a function of time <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_c=T_c(t),<\/span><\/span> we can find a differential equation that will allow us to determine the temperature of the coffee at any moment. Initially, we have:<\/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>Where <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> is a proportionality constant, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_a \\lt T_c(t)<\/span><\/span>, and the negative sign indicates that the temperature of the coffee is decreasing. Later, we will see that this equation has a solution of the form:<\/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>Where <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> is a constant to be determined.<\/p>\n<p><a name=\"LasLeyesDeNewton\"><\/a><\/p>\n<h3>Newton&#8217;s Laws<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=885s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Newton&#8217;s Second Law is, essentially, an ordinary differential equation,<\/span><\/strong><\/a> since in the expression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F=ma<\/span><\/span> (force equals mass times acceleration), the acceleration, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=d^2x(t)\/dt^2,<\/span><\/span>, is the second time derivative of the object&#8217;s position. Through this law, we can find relationships that describe the motion of bodies, which are actually differential equations. A simple example is the study of springs: if we have a spring attached to a fixed wall on one side and to a mass on the other at equilibrium position, and then we displace the mass a distance <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> from that position, according to Hooke&#8217;s Law the mass will feel a restoring force <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F=-kx<\/span><\/span>. Then, by Newton&#8217;s second law, we have:<\/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>Later, we will see that its solution is of the form:<\/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>Where <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> are constants that will be determined by the <strong>initial conditions of the problem<\/strong>.<\/p>\n<p><a name=\"ModeloDePoblaciones\"><\/a><\/p>\n<h3>Population Model<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=1184s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">The growth rate per individual<\/span><\/strong><\/a> of a population is equal to the difference between the birth and death rates, that is:<\/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>If the birth rate <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> remains constant over time and deaths are proportional to the population, that is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M=\\alpha^2 x(t),<\/span><\/span>, then the previous equation takes the form:<\/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>This is known as the <strong>\u00abLogistic Equation of Populations\u00bb<\/strong>. From this equation, a generalization can be constructed for many populations <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1(t), x_2(t), \\cdots, x_n(t)<\/span><\/span> that compete with each other to exist as follows:<\/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>With <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\in\\{1,\\cdots, n\\}<\/span><\/span>. This is what is known as the <strong>Lotka-Volterra Equations<\/strong>.<\/p>\n<h2>Conclusion<\/h2>\n<p>Throughout this introduction to Ordinary Differential Equations, we have explored how mathematics can precisely and elegantly capture the changes that occur in the natural world. From the cooling of a cup of coffee to the motion of a spring or the growth of a population, ODEs allow us to translate complex dynamics into understandable and analyzable mathematical relationships.<\/p>\n<p>Understanding the structure and meaning of these equations opens the door to multiple disciplines, such as physics, biology, economics, and engineering. This class lays the necessary conceptual foundations to continue with more advanced studies, where solution techniques, qualitative analysis, and numerical methods will be explored in greater depth. Most importantly, however, is having developed an initial intuition about how the language of change \u2014differential equations\u2014 allows us to describe, understand, and predict the behavior of dynamic systems.<\/p>\n<p>In the following classes, we will continue developing more powerful tools and applying them to new contexts. Differential equations not only offer us a way to analyze reality, but also to imagine how it could evolve under different conditions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Introduction to Ordinary Differential Equations This class offers a detailed exploration of the fundamental ideas that govern these equations and their applications in various fields. Beginning with an analysis of the nature of constant change in the world around us, basic concepts such as functions, derivatives, and their relationship with continuous and discrete change are [&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":3,"footnotes":""},"categories":[567,1150],"tags":[],"class_list":["post-32794","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","category-ordinary-differential-equations"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Introduction to Ordinary Differential Equations - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Discover how ordinary differential equations explain real-world phenomena such as motion, heat, or population growth.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/en\/introduction-to-ordinary-differential-equations\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Introduction to Ordinary Differential Equations\" \/>\n<meta property=\"og:description\" content=\"Discover how ordinary differential equations explain real-world phenomena such as motion, heat, or population growth.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/en\/introduction-to-ordinary-differential-equations\/\" \/>\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=\"2022-03-15T13:00:36+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-04-01T18:27:43+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=\"Introduction to Ordinary Differential Equations\" \/>\n<meta name=\"twitter:description\" content=\"Discover how ordinary differential equations explain real-world phenomena such as motion, heat, or population growth.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/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=\"6 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/introduction-to-ordinary-differential-equations\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/introduction-to-ordinary-differential-equations\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Introduction to Ordinary Differential Equations\",\"datePublished\":\"2022-03-15T13:00:36+00:00\",\"dateModified\":\"2025-04-01T18:27:43+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/introduction-to-ordinary-differential-equations\\\/\"},\"wordCount\":1275,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/introduction-to-ordinary-differential-equations\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/edointroflat.jpg\",\"articleSection\":[\"Mathematics\",\"Ordinary Differential Equations\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/introduction-to-ordinary-differential-equations\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/introduction-to-ordinary-differential-equations\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/introduction-to-ordinary-differential-equations\\\/\",\"name\":\"Introduction to Ordinary Differential Equations - 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