{"id":7362,"date":"2022-03-21T14:54:23","date_gmt":"2022-03-21T09:24:23","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=7362"},"modified":"2022-03-21T14:54:27","modified_gmt":"2022-03-21T09:24:27","slug":"taylor-series-method","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/taylor-series-method\/","title":{"rendered":"Taylor series method"},"content":{"rendered":"\n<p><strong><a href=\"https:\/\/math.iitm.ac.in\/public_html\/sryedida\/caimna\/ode\/taylorseries\/taylor.html\" target=\"_blank\" rel=\"noreferrer noopener\">Taylor series method<\/a> to solve the first order ordinary differential equation<\/strong>&#8211; The general first order differential equation<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/L02ZI7syoVYbVUDA0sI81Y-Ra4nt8Ythm7ANHtydr3VY3T6Xi03b5x17tg-gjZNMK_fVtJoj4cXcaM7Z9oSS77ljI2vufDWSxJx6YjhrQNFDSQ4nkL3Cs8r9Jh_SxyFnSPha8DSj\" alt=\"\" width=\"142\" height=\"36\" \/><\/figure>\n\n\n\n<p>With the initial condition y(x0) = y0 \u2026(2)<\/p>\n\n\n\n<p>In general, the solution of first order differential equation in one of the two forms:<\/p>\n\n\n\n<p>a) A series for y in terms of power of x, from which we can obtain the value of y by direct solution.<\/p>\n\n\n\n<p>b) &nbsp; A set of tabulated values of x and y.<\/p>\n\n\n\n<p>We solve case (a) by Taylor\u2019s Series or Picard method whereas case (b) by Euler\u2019s, Runge Kutta Methods etc.<\/p>\n\n\n\n<p>&nbsp;<strong>Taylor\u2019s Series Method:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <\/strong><strong><\/strong><\/p>\n\n\n\n<p>The general first order differential&nbsp; equation<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/UUMGGswsEsC3QqyetoWNTv9UPNqTG_mUuAr8Uq_7Sp6mEvoA5rQNPGekPqBwS0Dq1ozM_mvSdqwG2rKJr6OIw-QP9vaWx2UVGe-rGdqI8ygwpbn8EXBrD_Mw1D1Gi3ASvWStqdwh\" alt=\"\" width=\"141\" height=\"36\" \/><\/figure>\n\n\n\n<p>With the initial condition&nbsp; y(x0) = y0 \u2026(2)<\/p>\n\n\n\n<p>Let&nbsp; be the exact solution of equation (1), then the Taylor\u2019s series for&nbsp; around&nbsp; is given by<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/0zWRJa-0GjA5FkTUqNinuVOVuXQvISTjoaZagQjRWHbC2wIOet3aaTigqONgqtx_aPEv2rFx_3llUcA0vCkKMzUjClKE43MFyVh_yCkumvC238ySK1fFQAXaUJzJfwTpmXoX7Dph\" alt=\"\" width=\"315\" height=\"41\" \/><\/figure>\n\n\n\n<p>If the values of\u00a0 <img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/8q_LChjzwM0S216kWYg6YokplpmDzkVc9yhl9_2-_0MCdyt7cVC5lCEfpFRjFcfWgUrAY37BsN5MHbhmOdwW2BxqU5u1i8nvfG9bWyM2EpJDNLdtxpR2UcwsRWMqtsH5mzET31AH\" style=\"width: 100px\"> are known, then equation (3) gives apowwer series \u00a0 for y. By total derivatives \u00a0 we have<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/U0dh6Y993Aj9wPpjHKxQeYmEWfK4kDzSt1JHnbhkjeuHHUd1VKcTcKmPaPvDxrE9VSBu6mVIhdHGsP7ELo4tc0WKNfeQ3VLFrgsWitAmcqjWJHLVWEt3xARXDpcLv223RF3fS_P8\" alt=\"\" width=\"309\" height=\"96\" \/><\/figure>\n\n\n\n<p>And other higher derivatives of y.&nbsp; The method can easily&nbsp; be extended to &nbsp; simultaneous &nbsp; and higher \u2013order&nbsp; differential&nbsp; equations. In &nbsp; general,<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/yuW_PLKFaUWa1ErN50vm9NF_b0o98bifzwvbDV_NJ2nfY9lI6FdF_bADP5wuF5S4R0pnyhnFNV7LJZKDi38B5GbP2loNYu56mXoaDOIJrMhyrhyaubHcpb47rwnQtBE4PVS7QUOF\" alt=\"\" width=\"302\" height=\"41\" \/><\/figure>\n\n\n\n<p>Putting\u00a0 x = x0 and y = y0 in these above results, we can obtain the values of\u00a0 <img decoding=\"async\" style=\"width: 100px\" src=\"https:\/\/lh5.googleusercontent.com\/-ECik4UKSCjt3zlGX2stC---No7EhsvVotVGVdfLYRPMt_zoWxH3hz8KQJnuidL6Ps9SlzSZPjshzBWNaItsSYFF4iL4kk_qlZDT4aRtejm_ac1pZuX-ewPyslDfoOCZBRtq8RdZ\"> .finally, we substitute these values of <img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/hPMSEHlzOYS9P2E6g7d3yLcBTt4PWN0eiijPE02cwhvSP9s-iVfCSV7vayyCRgjvWlGfroIFNokre4lC2j3m3JJf91q4-9K9Vfee8eyxNC9n3k4waQ2sTmfMiu9-WqJTDd5mfkT2\" style=\"width: 100px\">\u00a0 in equation (2) and obtain the approximate value of y; i.e. the solutions of (1).<\/p>\n\n\n\n<p><strong>Example1<\/strong>: Solve, <img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/6Veug8Rstvhe9jqq23juuyfsJyTM0uXoFH2ArDtQ3H-Lf4Bmh1noWC--iJj81aLJYmaCN0HuuO3bDww7Jhvev_JUZ0yw5gh1HfAxnMrkC-8wysoGXYxtyeXn356F06D71--S3WMa\" style=\"width: 120px\">\u00a0 using Taylor\u2019s series method and compute y(0.1) and y(0.2).<\/p>\n\n\n\n<p>Sol:<\/p>\n\n\n\n<p>Here <img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/TQBoZgGeuQEijr6ILr-i9sZcSdfMV973b_40E-QDdcZEMLVwr07A9T9Q5GuzOc0l-Tg_hAXvdewuf6FnN-p81L9KvklET-rpF17kP28Iku3iHJO4AxqOQKCJVOG3LKC-B2NXW0NI\" style=\"width: 120px\">This implies that <img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/95NfrrNLCoo-VNkg8LdGy1A5zMy-2JicAy8CuYtp9G1G-n9SyFWe_7eqY7nP3mE_Z7s4fgczjOEOCAHED28kxH44yYpKn58xMKI5eNGQlsHVlUBUtV6SLvI5E1jMIRvwMjZ-06hA\" style=\"width: 120px\"><\/p>\n\n\n\n<p>Differentiating, we get<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/l7qxWbykdDV9UvqWahlse-ChAYUc3DAv3JSrxoJkWCKc9vrhIlMLaOVSi1Wl68uR_JeuH436TuMW0_eyg44368z7VQctJ9-w1Kkug6hzDaxukfyP5LNFrZZWYZi72qpOMZkqGRE6\" alt=\"\" width=\"436\" height=\"119\" \/><\/figure>\n\n\n\n<p>The &nbsp; Taylor\u2019s series at x = x0,<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/va0EtWNb9A3JEwiiiOOCNBq-A0PiBSGxRsPkefLyZLC1UyF75D17aY187e7y_JZO5Y3-byq2Hj05yOWrpNNyzNq5tIMrkFByI9dpb_6lUyzs59VN3_F0Xr1eNwaKEnUSwBGXUBXG\" alt=\"\" width=\"452\" height=\"122\" \/><\/figure>\n\n\n\n<p>At x = 0.1 in equation (1) we get<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/hB9ypUCSbK0Et68VKeA7WWQcC_jcnMita97LAPNOIPEYeYaSSj5JVRUbSMW2ZluuMvEol4XuXjGUxBHVy0OS02l8x0XHPintlbSI84vf7fSOE_udq2H8vZyvpbZy3J3Q4H75Cmdm\" alt=\"\" width=\"330\" height=\"63\" \/><\/figure>\n\n\n\n<p>At x = 0.2 in equation (1) we get<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/-exHjAgBp11pfSvBxlhP2lEscHOrCL2DXU5MRTJ_Jqpaqppj39DrhfsHgbWt-DXHh1XNe3H7xlO7Xx-xHAQ3rQ11dW1F7RqklS2_SCOqJC0mcfkQCCVnnMJikC4WTHRQF0I8T8hH\" alt=\"\" width=\"286\" height=\"61\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Taylor series method to solve the first order ordinary differential equation- The general first order differential equation<\/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-7362","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>Taylor series method - Goseeko blog<\/title>\n<meta name=\"description\" content=\"Taylor series method to solve the first order ordinary differential equation- The general first order differential equation\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.goseeko.com\/blog\/taylor-series-method\/\" \/>\n<meta property=\"og:locale\" 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