{"id":5700,"date":"2021-09-29T18:36:19","date_gmt":"2021-09-29T13:06:19","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5700"},"modified":"2021-11-21T20:15:58","modified_gmt":"2021-11-21T14:45:58","slug":"formation-of-differential-equation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/formation-of-differential-equation\/","title":{"rendered":"Formation of Differential Equation"},"content":{"rendered":"

Here you will learn formation of differential equation with examples.<\/p>\n

Let’s begin –<\/p>\n

Formation of Differential Equation<\/h2>\n

Algorithm<\/strong><\/p>\n

\n

1). Write the equation involving independent variable x (say), dependent variable y (say) and the arbitrary constants.<\/p>\n

2). Obtain the number of arbitrary constants in Step 1. Let there be n arbitrary constants.<\/p>\n

3). Differentiate the relation in step 1 n times with respect to x.<\/p>\n

4). Eliminate arbitrary constants with the help of n equations involving differential coefficients obtained in step 3 and an equation in step 1. The equation so obtained is the desired differential equation.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Form the differential equation of the family of curves represented by \\(c(y + c)^2\\) = \\(x^3\\) , where c is a parameter.<\/p>\n

Solution<\/span><\/strong> : The equation of the family of curves is \\(c(y + c)^2\\) = \\(x^3\\)                            ……….(i)<\/p>\n

Clearly, it is one parameter family of curves, so we shall get a differential equation of first order.<\/p>\n

Differentiating (i) with respect to x, we get<\/p>\n

2c(y + c) \\(dy\\over dx\\) = \\(3x^2\\)                          ………(ii)<\/p>\n

Dividing (i) by (ii), we get<\/p>\n

\\(c(c + y)^2\\over 2c(y + c){dy\\over dx}\\) = \\(x^3\\over 3x^2\\)<\/p>\n

\\(\\implies\\) y + c = \\(2x\\over 3\\)\\(dy\\over dx\\)<\/p>\n

\\(\\implies\\) c = \\(2x\\over 3\\)\\(dy\\over dx\\) – y<\/p>\n

Substituting the value of c in (i), we get<\/p>\n

(\\(2x\\over 3\\)\\(dy\\over dx\\) – y)\\(({2x\\over 3}{dy\\over dx})^2\\) = \\(x^3\\)<\/p>\n

\\(\\implies\\) \\(8\\over 27\\)\\(x({dy\\over dx})^3\\) – \\(4\\over 9\\)\\(({dy\\over dx})^2\\)y = x<\/p>\n

\\(\\implies\\) \\(8x({dy\\over dx})^3\\) – \\(12y({dy\\over dx})^2\\) = 27x<\/p>\n

This is the required differential equation of the curves represented by (i).<\/p>\n

Example<\/span><\/strong> : Form the differential equation representing the family of curves y = A cos(x + B), where A and B are parameter.<\/p>\n

Solution<\/span><\/strong> : The equation of the family of curves is y = A cos(x + B)                            ……….(i)<\/p>\n

This equation contains two arbitratry constants. So, let us differential it two times to obtain a differential equation of second order.<\/p>\n

Differentiating (i) with respect to x, we get<\/p>\n

\\(dy\\over dx\\) = -A sin(x + B)                          ………(ii)<\/p>\n

Differentiating (ii) with respect to x, we get<\/p>\n

\\(d^2y\\over dx^2\\) = -A cos(x + B)<\/p>\n

\\(\\implies\\)  \\(d^2y\\over dx^2\\) = -y             [Using (i)]<\/p>\n

\\(\\implies\\)  \\(d^2y\\over dx^2\\) + y  = 0, which is the required differential equation of the given family of curves.                <\/p>\n\n\n

\n
Next – Solution of Differential Equation<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Degree and Order of Differential Equation<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn formation of differential equation with examples. Let’s begin – Formation of Differential Equation Algorithm 1). Write the equation involving independent variable x (say), dependent variable y (say) and the arbitrary constants. 2). Obtain the number of arbitrary constants in Step 1. Let there be n arbitrary constants. 3). Differentiate the relation …<\/p>\n

Formation of Differential Equation<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[38],"tags":[251,262],"yoast_head":"\nFormation of Differential Equation - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn formation of differential equation with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/formation-of-differential-equation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formation of Differential Equation - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn formation of differential equation with examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/formation-of-differential-equation\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-09-29T13:06:19+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-21T14:45:58+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/formation-of-differential-equation\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/formation-of-differential-equation\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Formation of Differential Equation\",\"datePublished\":\"2021-09-29T13:06:19+00:00\",\"dateModified\":\"2021-11-21T14:45:58+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/formation-of-differential-equation\/\"},\"wordCount\":457,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"differential equation\",\"formation of differential equation\"],\"articleSection\":[\"Differential Equations\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/formation-of-differential-equation\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/formation-of-differential-equation\/\",\"url\":\"https:\/\/mathemerize.com\/formation-of-differential-equation\/\",\"name\":\"Formation of Differential Equation - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-09-29T13:06:19+00:00\",\"dateModified\":\"2021-11-21T14:45:58+00:00\",\"description\":\"In this post you will learn formation of differential equation with examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/formation-of-differential-equation\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/formation-of-differential-equation\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/formation-of-differential-equation\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Formation of Differential Equation\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - Study Math Online\",\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathemerize.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathemerize.com\/#organization\",\"name\":\"Mathemerize\",\"url\":\"https:\/\/mathemerize.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"width\":140,\"height\":96,\"caption\":\"Mathemerize\"},\"image\":{\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.instagram.com\/mathemerize\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\",\"name\":\"mathemerize\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"caption\":\"mathemerize\"},\"sameAs\":[\"https:\/\/mathemerize.com\"],\"url\":\"https:\/\/mathemerize.com\/author\/mathemerize\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Formation of Differential Equation - Mathemerize","description":"In this post you will learn formation of differential equation with examples.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathemerize.com\/formation-of-differential-equation\/","og_locale":"en_US","og_type":"article","og_title":"Formation of Differential Equation - Mathemerize","og_description":"In this post you will learn formation of differential equation with examples.","og_url":"https:\/\/mathemerize.com\/formation-of-differential-equation\/","og_site_name":"Mathemerize","article_published_time":"2021-09-29T13:06:19+00:00","article_modified_time":"2021-11-21T14:45:58+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/formation-of-differential-equation\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/formation-of-differential-equation\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"Formation of Differential Equation","datePublished":"2021-09-29T13:06:19+00:00","dateModified":"2021-11-21T14:45:58+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/formation-of-differential-equation\/"},"wordCount":457,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"keywords":["differential equation","formation of differential equation"],"articleSection":["Differential Equations"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/formation-of-differential-equation\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/formation-of-differential-equation\/","url":"https:\/\/mathemerize.com\/formation-of-differential-equation\/","name":"Formation of Differential Equation - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-09-29T13:06:19+00:00","dateModified":"2021-11-21T14:45:58+00:00","description":"In this post you will learn formation of differential equation with examples.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/formation-of-differential-equation\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/formation-of-differential-equation\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/formation-of-differential-equation\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"Formation of Differential Equation"}]},{"@type":"WebSite","@id":"https:\/\/mathemerize.com\/#website","url":"https:\/\/mathemerize.com\/","name":"Mathemerize","description":"Maths Tutorials - Study Math Online","publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathemerize.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/mathemerize.com\/#organization","name":"Mathemerize","url":"https:\/\/mathemerize.com\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/","url":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","contentUrl":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","width":140,"height":96,"caption":"Mathemerize"},"image":{"@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.instagram.com\/mathemerize\/"]},{"@type":"Person","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df","name":"mathemerize","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","caption":"mathemerize"},"sameAs":["https:\/\/mathemerize.com"],"url":"https:\/\/mathemerize.com\/author\/mathemerize\/"}]}},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/5700"}],"collection":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/comments?post=5700"}],"version-history":[{"count":8,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/5700\/revisions"}],"predecessor-version":[{"id":5795,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/5700\/revisions\/5795"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=5700"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=5700"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=5700"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}