{"id":2930,"date":"2021-09-23T08:20:27","date_gmt":"2021-09-23T08:20:27","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=2930"},"modified":"2025-11-12T09:35:08","modified_gmt":"2025-11-12T04:05:08","slug":"__trashed-34__trashed","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/__trashed-34__trashed\/","title":{"rendered":"What are linear differential equations?"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Overview<\/strong><\/h2>\n\n\n\n<p>An equation consisting of a differential coefficient is called a differential equation.<\/p>\n\n\n\n<p>LDE has many applications in engineering problems.<\/p>\n\n\n\n<p>For example-<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/N0Et9ywHy3-S5HKkW8KyrEKITTQqUDRw_XQVfxKZ6YmslOEnV2ZXvlQfuoAhvablKeDgEPym2Ks6TCNOhXf0P1tOCqweCq-coXDeBClWZLBtux8GDx2dpEopdRFwDLbFbWZMUMT3\" width=\"78\" height=\"44\">is the differential equation.<\/p>\n\n\n\n<p>&nbsp;A differential equation of the form<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/WFLAFrypcs5_Kp41NwEj9Fur1wQCOfJqm5RU3f0sj3aaRTmi4bSrF5AOgfh8doTTP_E50kSbwBHs-VVu_lGI4FOx1BQb8TNG99omTQ4ThnwOmo3qUsHQSO5FyZKSde82JNdPdsp4\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>is known as a linear differential equation or simply LDE.<\/p>\n\n\n\n<p><em>P <\/em>and <em>Q<\/em>, are functions of <em>x <\/em>or constants.<\/p>\n\n\n\n<p>For example-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/LuskcxBjb3hUf9rUS-rDgg7C5oKczRo2pz8z_Mjoao3yvbCvaK4Fwvv_7nl6T0Q8lKjKBvri1s4wjukxFrEG_7okSCKb28M0U_Wz4Fm3qnPcLGWUFHJvo3ywZJDTXxstwASVIdaz\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>The solution of LDE is-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/qXUhnHp0Q8tBW-jSZ0557S3UcmULexMw_JXdH7InbEwlcmQEq3n9fZq4Rq2u9E_pPMzEu01MQQsMtA2ZdRTMpPzidLHC4iJ-gutqYt7qNVlOef91p2iBy6Ir3375gcTKpYqKYIuh\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Note<\/strong><\/p>\n\n\n\n<p>1.&nbsp; &nbsp; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/ppwFpNnMweiDh5SOI2GL7W9o7ZAdjM0hBe2fwQYEaiLMyGI1p82TsMQAGZpXeHJ5E8PGyrLdcsOrQUeNkQ8LmR8QnWW_NujNXYrOot7PYeDxC0qbvko6Yx5g3PTSEXd5Tpwb4TPD\" width=\"203\" height=\"26\"><\/p>\n\n\n\n<p>2. &nbsp; &nbsp; If the RHS of LDE is zero for all x then it is said to be homogeneous, otherwise non-homogeneous.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Solving a linear differential equation<\/strong><\/h2>\n\n\n\n<p>1. First change the given equation to the standard form of LDE, which is<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/mJfGwQ0F5kyjCS4m1gXxTM6CXX-QaDZWLWGXO7z-9d1oXx_DaLLhKyyRwNGEO-J0RNpRixTdbR4QFtF1m3zctDJbZhB2tT9-6dgei1ROvrY5LN_J64LQR3fKBx2drHF4X3ovD_gf\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>2. Find the integrating factor<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/uFCdM5C2SfWtNGXRos0lU6wC49JewmhmM4ZsUD2P-jhu4wh5N2EPUQ7srucVM1jnJgeu2N_4MG_BSy4t3xSb8v0nxwA0BG5WMbU8y5O0FgUY91VFWM5DKLyusrvf63acNmZYos3l\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>3. Then the solution of LDE is-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/-97aQStwvB7548hHLWOuRtpU6XHb5Et5yJPBbiJ6qMyrEfV68ippplAgRtRwlA7AOoUT-c0oBOnqhhYYiYKiNRPOE4sTUexk3ZsSf0FPtGzWUyjyON4yO6cxK9YKOPpvFOVKr4-h\" alt=\"\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Solved examples<\/strong><\/h2>\n\n\n\n<p><strong>Example: Solve<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/zk260th0OrPf8s9Hfx-dAZ_-TYbZtUUaq5tMi6DQjqGfaKIxld1n-M5KD2HVEo53kXgT1F1U9Vcs2metLL6dh_Xvavi_ka9dDtM4O03x4VcaK4nw-twAemjxG5fPlqrhWRX-ksC_\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Solution-<\/p>\n\n\n\n<p>First we will convert the given equation in standard LDE form-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/hJgFktWob_nUB34oBA-kMvw5HzG9rsRnAqnm3necT9n-9WSIYHylF-2C1mSAAgtZxVaqwNITUPjMkUFXR3vvc7qmqCtv6_dvQ9AET2jhT6ELgv3bRbJlGn0C6hdf-kZcqnZXFtHv\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Where Q = sin x and P = 2\/x<\/p>\n\n\n\n<p>Now we will find the integrating factor-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/uFJd8wPHmGyorqTT79xhRDMwyscbK72dLNNRUihYdrsaV4Ph5m76RLPH9bvaTypdcmFcvREFeOu3eruMi1bV7sd_OmpiQTzVBsnbhA1vW5kqz_0qcpLsadjrgc6iaEb6zXl_g94K\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Then the solution is-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/fgQMZGr6ZX_tA4bztHuqQTyeEUos7rD2fUa6CEhgFQFfQBR30-zB7zPt7F1tl2-u2p0dugWALCYaLrrqfAA8MbLBsLOedcL-V_LIlY0qbGO3yGkqZDXY61l6ZLTVk_IC6IiUkbBR\" alt=\"\"\/><\/figure><\/div>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/I2db_nex4Vq4sK1j0Z-CWzqc8wikdvDxEevk_RPaMvt0QGNxrsMGooNnueniQSH2fToju8fLN4qQKqxhivTMocIX9P0pU_meyp5RhRoDw65IA6eUnVc5XC32mKVInDz9n4fObfHx\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Integrating by parts-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/G-SpPW7fZ96OgzYJWGHDTrSxEJv5adzxL_iQyZPNhpmlh0gEaXco70pB-My7m3C1ZreH_0mdHK4UpqyIyjOYEqEjI9Y7KRJ_of3eoiJwMUNjMY9e2rXFKn2sL-wWdbeiI1zwmmXi\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Example: Solve-<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/QEhqQzmxQdLY-yoMOgWKr_UzpoFRv9aHXiMitfjQzMe5MXuR1TTfwF5rgzLxVyYHUoLtOWbMyHqX9MmE8dVzDrQVB6_nE6Ev1Fa9xDE3nd37dYm7Fpi4DbOT8ruj6_DaZ_d-KhLa\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Solution-<\/p>\n\n\n\n<p>The given equation is already in the form of standard LDE.<a href=\"https:\/\/julichain.com\/careers\/\" style=\"color: #000000; text-decoration: none;\">https:\/\/julichain.com\/careers\/<a><\/p>\n\n\n\n<p>Now we will find the IF-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/2xSLX5MVGnm8e1tytxkYEc9cbJVPPqAFkksGVcD2wR8T26l40vsVXRGrn6qFz_Xe4ZnvSmynuYO6ozP0nVmr9i4JIpZ78x-9eU2tVVHzb2225OK7I991xsXvHpdjlICqZqwC34s4\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>So that the solution is-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/PLlRsT2CiD7H6etzk8zVwhGiFogtqi16fYbwIJQvsc1UGRxyoA5uQwePhdvCRLa9gBIDDQyhsI5oaZNaj7Dl26nobkm8kVP7VQ_j150q0mHJ1bbDbKg1chPBCl_lvKzPRpLN5Phu\" alt=\"\"\/><\/figure><\/div>","protected":false},"excerpt":{"rendered":"<p>An equation consisting of a differential coefficient is called a differential equation.<\/p>\n","protected":false},"author":3,"featured_media":2994,"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-2930","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What are linear differential equations? 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