{"id":2917,"date":"2021-07-12T21:06:37","date_gmt":"2021-07-12T15:36:37","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=2917"},"modified":"2026-01-12T16:23:27","modified_gmt":"2026-01-12T10:53:27","slug":"what-are-homogeneous-differential-equations","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-are-homogeneous-differential-equations\/","title":{"rendered":"What are homogeneous differential equations?"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Overvie<\/strong>w<\/h2>\n\n\n\n<p>The differential equations having function of the same degree are said to be  homogeneous differential equations.<\/p>\n\n\n\n<p>Differential equation is an equation involving an unknown function and its derivatives.<\/p>\n\n\n\n<p>Ex. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/3esIioplHGtx-vkcRkeY_UsMEi2I3usEApQNiYRsVgkhIy2cCQspDv6gDg0Ofn3joeU7tAkPjjexmBkzH6fgaDUY5ueS0FeBUbgUbNBJ7ZUDdUR-73YfgOiOGt9Ndx7bcRzW4rwK\" width=\"98\" height=\"32\"><\/p>\n\n\n\n<p>\u201cA differential equation is an ordinary differential equation if the unknown function depends on only one independent variable.\u201d<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Notation<\/strong><\/h2>\n\n\n\n<p>The expression y\u2019, y\u2019\u2019 , y\u2019\u2019\u2019 \u2026\u2026.y\u207f are used to represent derivatives.<\/p>\n\n\n\n<p>Here,&nbsp; y\u2019 = &nbsp; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/zt8EsIIOQOtHs1dd2l46Fge-1WWunnG4oIrbRhuT9ZxmK-NiqcUBlR7c1HyCG63iCutmZfr6F-xh3j71vsMdQ9vuS9EBg51f29TbbJe8yDI9HLrhSNfoIDDA_wtAAS7uFov-0bam\" width=\"29\" height=\"34\">&nbsp; &nbsp; , &nbsp; &nbsp; y\u2019\u2019 = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/UosEVfJ4XNOVSvyC6fek98A-OhxqigJ2mJ-PDuK3Nq6rWJPg6AyRrQFH9kRkdgNJXLCUotlqc_05CgKTqQ_bo5wtgR4ZGeJNc23XwA4ZR7o-hXaR1VT4Gn8yhCFtRQQVXzzo_OHz\" width=\"29\" height=\"36\">&nbsp; &nbsp; &nbsp; and so on.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Homogeneous differential equations<\/strong><\/h2>\n\n\n\n<p>A first order differential equation is homogeneous. It can be written in the form as,<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/CMVb-vtvpCqbLxue4DKoD8gQuZcLMSOUCKGjRsnqNCpmU-7gkvBeN1qx3yX8a3Sb2TmOhnaO7UZ6hwnQif3BPKLa22RYYjEx_VQaGcnqx0DZgf5V9NmCvDZjJVm6JwHMc2pdPSVl\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>And each term of&nbsp; and&nbsp; has same degree.<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Homogeneous Differential Equations\" width=\"1170\" height=\"658\" src=\"https:\/\/www.youtube.com\/embed\/ZEJVyybsiT4?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Solving a homogeneous differential equation<\/strong><\/h2>\n\n\n\n<p>If the differential equation is homogeneous, we put<\/p>\n\n\n\n<p class=\"has-text-align-center\">y = vx, and dy\/dx = v + x dv\/dx<\/p>\n\n\n\n<p>We follow the Steps given below to solve a homogeneous differential equation-<\/p>\n\n\n\n<p>1. Put y = vx and dy\/dx = v + x dv\/dx<\/p>\n\n\n\n<p>2. Separate the variable<\/p>\n\n\n\n<p>3. Integrate on both sides<\/p>\n\n\n\n<p>4. Now put y = v\/x and solve.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Solved examples<\/strong><\/h2>\n\n\n\n<p><strong>Example:&nbsp; solve the homogeneous differential equation given below:<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/UYgckh8l-WpO36u6TohHalNia2LL9eYY5QOV_fu8tG-BI5ShAIVPSiNc8HT_x6kMc3KWX3PzlCjfFPlfTpcbKvRSxlXx_JtDTrogkp2_uiH9ievTbKS0DhH5F33WzJFpIX6vlBmN\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Solution:<\/p>\n\n\n\n<p>&nbsp;Here the function is &#8211;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/F8FgIHXF49z4c5-OgktoX-Z0-VCWwwZpEaUinHlr7N4tswA-mPmTjBFeHgcD9-oBTUm-fBEqmp2YuuIshG6p2uOfbn7_uE9kFdauIsrNnLQWs-CKdq8NzpbfFGF-scCkstIGhmse\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>&nbsp;We will put&nbsp; y = vx as mentioned above &#8211; <\/p>\n\n\n\n<p class=\"has-text-align-center\">dy\/dx = v + x dv\/dx<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/37lyQOUZ9Pn-GWiJq595zkL0K09UXZA3YhAXTejt4S07CqH5HRpANKVR9173VbTamvIoWdLu84TOey7fiBv16pteqJpuii0g-KwVdbi9aP5frLKI7-HAWR6f7WmxPQDiCxFC6wq3\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>&nbsp;Separate the variables &#8211;<\/p>\n\n\n\n<p class=\"has-text-align-center\">dv \/ sin v = dx<\/p>\n\n\n\n<p class=\"has-text-align-center\">cosec v dv = dx<\/p>\n\n\n\n<p>On integrating &#8211;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/YsxQcBo8y2oivLpZZZpnal-vOnsDQchfjq_huY11E58U4NQLSwFqQjA1Hf6L3ZkAT0KlZ9rzamailg74r9P9uUeKzTTc_eTXVB3sjNe-Bhs0I0yXEDHO_Rox-bRb_tZK0QMNoZAF\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Example: Solve the following homogeneous differential equation-<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/HBXGC_ND1uRKnzGePHG0IcL_DFNOMpjV1ZeJly8wfK7qcUPYQ7AorL6ee0mA3vc8Dz-vCipgYq1YmetlRQkvczuUN8E6rkDnVP5qjToCMPXGDsHtBI8Fatc3qaucvB-vYdAV53oI\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>On solving, we get-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/7TBTITG_0pnwyBvcSgxPSpHaMLfz4ySF05sKsgltZ5BaI9RykOFIsmDs4tpci3GnWC9L1wjrDGzA9cEkkv6pDrQhYGIDeJSTim2j8G_lzFqHlacfx-AKbKwJX5n0Y2O9LCw3TcUv\" alt=\"\"\/><\/figure><\/div>\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/e45m5CMLg0wB-kAasaNKd-K-5Px44YnvcQ5DnejXF4Gxeon5822LPmwRaafDmjW79lMadW1QB-jN6UEF-TboCyIvjS02NOh5L_lby3Lnwtd3-GI9WS-WsdSvML1xCD3fXOCYjPts\" alt=\"\"\/><\/figure>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/R0j4K5uvpdP3NjBfjxNHj__Tc7Y_PnLBl96ZNqCLmHqSs9RnAGWBald6kAVzfROdHHGAMxgLoiC0psZtteiBskG218PrfW6gxKv6lMgnWyYU2BOgf3Mz7NZ_8BCli2ThkcjDUT-z\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Now separating the variable, we get-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/s4c7KOm8q-MV9b6OukrUKLsLoxt_AjUNWhy5toOnI2wJH9cY_oRUy6qIS_WyAFmGEuDjHK55DpPj5XWh2rvz3AXV74GsufNkRtsTiUU5QEQ_MQNdpg2CGy8bBAeg2rO0W8quZHzU\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>On&nbsp;integrating both sides, we obtain &#8211; <\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/0lDUoN8NCUqlgH4PxX2ZtIM8jVQfrPZ81u4buN7bfKktPU5tZholnSBSlaIxnbA50gCaWv4WEfLlfD2_MMDk_wEAi8rFbahIvtdBL8lfKmFv_119uIgaWGPUvjNbIhVj5bqQNhOm\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Or<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/yu5GGpRFV2xeXCd6zgTHE0S05AKBGbIIkkQ5-raCxqNmIhC0whoNV6I3V8EAhfUWwAapGeDSHxtjN1Ff8plPaLY20bLGK3BwHBBVw-csjuLjZxGeU7FU_CnrT3qJkBZ7du5Ck4_C\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Interested in learning about similar topics? Here are a few hand-picked blogs for you!<br><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-cayeley-hamilton-theorem\" target=\"_blank\" rel=\"noreferrer noopener\">What is Cayeley-Hamilton theorem?<\/a> <a href=\"https:\/\/jhanpa.lgoms.org\/\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/jhanpa.lgoms.org\/<\/a><\/li>\n<\/ul>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The differential equations having function of the same degree are said to be  homogeneous differential equations.<\/p>\n","protected":false},"author":3,"featured_media":3483,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31],"tags":[],"class_list":["post-2917","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with 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