{"id":5182,"date":"2021-09-23T07:28:19","date_gmt":"2021-09-23T07:28:19","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5182"},"modified":"2021-10-30T07:40:29","modified_gmt":"2021-10-30T07:40:29","slug":"__trashed-9__trashed","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/__trashed-9__trashed\/","title":{"rendered":"What is method of variation of parameters?"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Overview<\/strong><\/h2>\n\n\n\n<p>The method of <a href=\"https:\/\/www.britannica.com\/science\/variation-of-parameters#:~:text=Variation%20of%20parameters%2C%20general%20method,differential%20equation%20will%20be%20satisfied.\" target=\"_blank\" rel=\"noreferrer noopener\">variation of parameters<\/a> is the general method which we use to find out a particular solution of a differential equation by replacing the constants in the solution of the homogeneous differential equation by functions and evaluating these functions so that the original DE will be satisfied.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Method of variation of parameters<\/strong><\/h2>\n\n\n\n<p>Consider a second order LDE with constant coefficients given by<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/7xuCZBW37MeOpfL0ThvBN875ofjIv6aFx_nleONX7t6_Luj7EunNnqZDp8ddt_5c_iGzQUOn_IzRmcmekWU0FnLp59V1eOWnv0WQrlJJ0cTGf7M9ZcRNlpHg74_3o7FJ9itJjR8F=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Then let the complementary function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/DIwgxbMOGClGLiEpdH2JNBaIhXprsP2HZxL1zVRa1Q4eMjTmUbDbpHrJY78lqvTuig8S49jlKgDsW22oQ3dij_118xzfrNptbZPePFAm1OxvZaAWxttJPwAP-r5-e4TSG6hwfRco=s0\" width=\"18\" height=\"23\"> is given by<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/2aIRWzsTy66Iv17LUftz1in0qZLKBlKHFIFbR_x_6SXjzJs9zBtf7eE7qZKf1e0eBp5k7eTs3RzLiP0So8Bpjk-sbFZ4pD_o8Jr86TJAl_aglt8UHEv5mPOPbXTm6jqGtAqRthO8=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Then the particular integral is<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/_l8vyJc9e34kUZLWMwTFNWM2ZDaCmIJ6OnbWlv5B5s3k-C9_1THDz72sxS6rQNdMS2iZ0Rm8cSqqQiza_9y6fQvFVnidG7WOyqI272KTyJv4QCUZZ-g7hu0hQhSMjwtQz7nXs4pj=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Where u and v are unknown and to be calculated using the formula<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/1OkHIDt2-l4j3U5jySGNK15GISaIXbugXwEJMULI6BrS3HmxLjClcpnGfvU3WYQvID6HIWk6mpoXRaHT1MVIFRbAt-7zdpFyg5qgHu2JjGfzRTUgD-pGqoSEERnHEq-QVdBM-470=s0\" alt=\"\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Solved examples<\/h2>\n\n\n\n<p><strong>Example-1: Solve the following DE by using variation of parameters-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/UkWi3DQ9IXQHOA4virP-9NUw6VrDeU-LGHnbb5xeHfncjMwCfB7Q1VanUt09Wt-CRP5ELIu3zBd-tJr6RWD3nPyt27lbfdXN-DnGcvf1P5v7Kf7Tt5y1iOXKaHGkdex08BBrI0Ft=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Sol. We can write the given equation in symbolic form as-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/vD5dEWubcIzCfR03u29HYE8oxl5CTBTZ9UJ2UCX21uGs8CLwOylLUw-XestC-7qAxdHvrQe_4d6vq0vvbuTgG8ycIx-PGtqiPxKWjwel3FSuh5KpiEz1GLV76VOLy-HJmqVsfEQ1=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>To find CF-<\/strong><\/p>\n\n\n\n<p>It\u2019s AE is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/IkXYllypnxLO6OEEkxxcaznshnTbPBndrLsBT28RnViG10sWhRYMZdfb5twKyVNtorI-KUvVQHFSmSCHLmX-jzqIDAjlpGRzPC_fM0K2uCj7fB3PKJ6D8IJnTSoVkMwLkAF8uoRg=s0\" width=\"176\" height=\"25\"><\/p>\n\n\n\n<p>So that CF is &#8211; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/5AZ07jp02OK4eRh-zEt26YeKRAyBedEoFpmdDS8ZAS6Aqrls7dMmdPBW60DzfR-QKtXLbizDiXDU85t0A0ids21u_niz0GuTqv0LWp76uf7hBomFacGyGsQPfmvIEw9794ua14Q3=s0\" width=\"177\" height=\"23\"><\/p>\n\n\n\n<p><strong>To find PI-<\/strong><\/p>\n\n\n\n<p>Here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/hz1QgapslH7Bo3OIhdpoVJzImWuR9cw4VL3s7p9R6UYbJgKgTazWwFJ2HehqP1_wuDe5jAteL2ZwPRhjw0jDU-bMtml6jjvtv_6N_RA_u4mzaSVRicBEzkgsQZjr3UAE_fuCECZf=s0\" width=\"284\" height=\"19\"><\/p>\n\n\n\n<p>Now<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/X68i8pQuytH7rsC1n_myBwS7UZUXoihNFsa_a4Qr0cY3Ey8IOMIKwhz2-51JpX5zmJT3EW5UYb4hwx0O7cdRCDLKq5StFnRbEjuyfT1gqYBYNlJYk9NLp5Xx4tVYXwMdkmzZ9_YT=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Thus PI will be<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/uGCXnAyboUi4jBElhkKQL_GQUcwO5n8z96j--2cdupP-8V1bLuDJt5FK8sQpurAirTETPDpVaDq1v4nHaN3GH_odR2fxBOzfAr7hgxAw1n6MRNxfHxbvamiwsYh0P-S9_XDmBQdW=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>So that the complete solution is-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/EQxdAmjYgsPSwBkhpIQ-fE0lEQp90PXGSvxl2UZ9WnJ9m-H1olpd6IsMMWrj6QADpOlX-NSbUKiF22Bbyn0Stw4fRsrTfN09F2k_y5T9N8h5wJBz8wI-8gJ0uCgy9pWdRQidMpjF=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Example-2: Solve the following by using the method of variation of parameters.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/JZevz7e9mpZHezeOaKJfmA9KXWASCYJepUljgP57jAglzWPS3sOelYxUpggjZB6rz7tAhc0StBeHx6ZX2dWfZMTIO-Ku5WtVKLNOMkPrrsuvZDNYdx-z0L-aK9o4vUTnql5Y7plY=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Sol. This can be written as-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/4gTJuD7IVuHRFi4hN7eZnmppIH02IPymUcCplS-6YXJckHnWa71YgE5aLX6ZZbXQ5j6_RMvb0OkTaXqBpxyE7L587_QmDU7yN7E2l0AWHjsH_c0E1Byx1kBoxiPpjv9_v6gMuuGB=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>C.F.-<\/strong><\/p>\n\n\n\n<p>Auxiliary equation is- <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/qqYZvx_Ja94hy1JKusrmNLcSlJwMMe-GOw80ewIfUmzU6Mdd2IFMwsO_FBorcRZUB1JFWrtQ5CsDv6uwJ28lLFs2k7EwglJ-3uKo28h4aoM3bTOmtgo551yeiy7PdfnC7Tapkgwy=s0\" width=\"314\" height=\"23\"><\/p>\n\n\n\n<p>So that the C.F. will be- <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/Hs-eqZ2IboswIad8UkCPaKqhz9wkqg-MVxOOdN8RYmNeafEWbUCssh_XWbXJwofEOBJDP7XkGUatdIWRc_f-jJDGzTCOmnyZzmRmAC25D6_QLG2RlH6Kv4i9BGzreiGlxjNOzWxN=s0\" width=\"91\" height=\"20\"><\/p>\n\n\n\n<p><strong>P.I.-<\/strong><\/p>\n\n\n\n<p>Here- <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/YMKL8z-ctRefrhjCpSXoCq58O_d83M8oruA4BV_zdBQQoIyJexDkw9YrQhQNXI26TK1WaoEyy0B2Ljit5B6zzcqFyGFhrrobfkHz6vNkpTE51sFwxRzntFCdun4mu9t5aZqiKxLV=s0\" width=\"247\" height=\"23\"><\/p>\n\n\n\n<p>Now&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/jlck8uNjOE0wj9R7_m8tQoX4z4qN7LznM-lLraNokU1tkPP4hdBZPAYIafIwFXQJmTCPeL42WI4WVG1HLTEZuLWPCyIrYY1cYqH7SypA_3if1xmSii6oQ8FRWdAbjtR5SKrxVbf5=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Thus&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/tPxkiJx2Vai2a-Rz1vLKTj8B-ss-znWjxM8dOsyIsMXIZ3If67EcOFjmvgEvqRCHXbv65ZzZXe4sJN5C4Ic_JJkkhARcX8HS1zhHyJ_iHCfbO_v4O5WN-YA10oCFbLNEfo6Pj5-7=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>So that the complete solution is-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/4hLeuuGY6XdlzXmrriCUHDz3yByUr6hiWsfjbnt623L7BBR9HJ-eG8bfs3holusPfR_6XdX42UJEFnWk6Ta1b5p4AvdriSebONXucLCtmXdezZVc8ukXIa3tAZX8DLhWEXDxGsvp=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-the-periodic-function\/\" target=\"_blank\" rel=\"noreferrer noopener\">What is periodic function?<\/a><\/li><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-a-linear-differential-equation\/\" target=\"_blank\" rel=\"noreferrer noopener\">Linear differential equation.<\/a><\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The method of variation of parameters is the general method which we use to find out a particular solution of a differential equation by replacing the constants in the solution of the homogeneous differential equation by functions and evaluating these functions so that the original DE will be satisfied.<\/p>\n","protected":false},"author":28,"featured_media":5516,"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-5182","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 is method of variation of parameters? - Goseeko blog<\/title>\n<meta name=\"description\" content=\"The method of variation of parameters is the general method which we use to find out a particular solution of a differential equation by replacing the constants in the solution of the homogeneous differential equation by functions and evaluating these functions so that the original DE will be satisfied.\" \/>\n<meta name=\"robots\" content=\"index, follow, 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blog","isPartOf":{"@id":"https:\/\/www.goseeko.com\/blog\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.goseeko.com\/blog\/__trashed-9__trashed\/#primaryimage"},"image":{"@id":"https:\/\/www.goseeko.com\/blog\/__trashed-9__trashed\/#primaryimage"},"thumbnailUrl":"https:\/\/i0.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2021\/08\/pexels-karolina-grabowska-6256066-scaled-1.jpg?fit=2560%2C1707&ssl=1","datePublished":"2021-09-23T07:28:19+00:00","dateModified":"2021-10-30T07:40:29+00:00","description":"The method of variation of parameters is the general method which we use to find out a particular solution of a differential equation by replacing the constants in the solution of the homogeneous differential equation by functions and evaluating these functions so that the original DE will be 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