{"id":5127,"date":"2021-08-31T11:24:43","date_gmt":"2021-08-31T05:54:43","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5127"},"modified":"2025-06-02T20:59:10","modified_gmt":"2025-06-02T15:29:10","slug":"what-is-eulers-modified-method","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-eulers-modified-method\/","title":{"rendered":"What is Euler\u2019s modified method?"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Overview<\/strong><\/h2>\n\n\n\n<p>This method was given by Leonhard Euler. <a href=\"https:\/\/atozmath.com\/example\/CONM\/RungeKutta.aspx?he=e&amp;q=meuler\" target=\"_blank\" rel=\"noreferrer noopener\">Euler\u2019s method<\/a> is the first order numerical methods for solving ordinary differential equations with given initial value.<\/p>\n\n\n\n<p>It is the basic explicit method for numerical integration of the ODE\u2019s.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Euler method<\/strong><\/h2>\n\n\n\n<p>The general first order differential equation&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/aB2cCe-xzgxptIHRtU_RVCHlzMBu_QFmCNmTSJdJnwsytK-TIq3mOKfhE0fypdtd9DGfiHcbCxzLKWy76W4lMwRB3GU9u1olGTjIcqG4SWOSOgGXyxOmeHm-rzILajI34oZTQKQu=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>With the initial condition <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/6cgDfskduZU0dNHKLLfD3hwo1C5-1_twRo_Qketlpk3_TYJKu85p_LEPJx4uEXvLWafC7nSDdTPzs4PW4ldm5U5T30kLicYhWlLWMyKTAg_CLonCwcEpYjr5h8Pbmn5Hci5FCLlg=s0\" width=\"78\" height=\"19\"><\/p>\n\n\n\n<p>In this method the solution is in the form of tabulated values.<\/p>\n\n\n\n<p>Integrating both sides of the equation (i) we get<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/KUXTBsX-zblkbkbrDEpsZE5Q2bo05cT7eWuubOHRoFw1-_4Kfti3xYfkwQAdARC3nhzoF2W4eTUqjZCKcb8RpP0UGb1T6xN72QGXf6BZFHWdqkIzcRRS-b2IPt-0W36peC6g5r5d=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Assuming that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/LciN-OdTevGwTnRtxKWRV2MG33cQBkZ5kNypl6ictnTMggOuQlDEShWYaB2i-DpoUzjaAZKd1yy086wS1uOqn20gzZMWDvkRJTH-SG4mG9gv3rQgoejyIBxZZ9kZeeZpSzgugax3=s0\" width=\"132\" height=\"22\">in&nbsp; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/snY75y1UeJTKtjhdLZmyG4Z_kWmIGnDhSIvvgVhOOucwNx59Sfy8T-QmWx3y1fjM42-woxHg0V8uls4hIgL3twZI4nfDG5idjrV48t0YCDtz8tq7zaLKJ_Y5UYh0LD0c8sUn530T=s0\" width=\"87\" height=\"21\"> ,this&nbsp; gives Euler\u2019s formula<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/4HJiwEh5bd5J-2grVYS_F-NEVS580ptZUtBlOrFarS5Mce8LCxTQIDRgO-eDOtDWjGz7T4wMsMVt-s3jCQt7AL1OZHh-HZcKCvaMqvmCNShqPUzHIggNImI5ntKt8w-2lxHQMZZ7=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>In general formula<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/rEAxBYfW6ayHkcK4YVZtYH48c45HDWRifdU63XQbT6bpl_qNN-YqpfGuux7DeTQYaptMz04mOnlDVM9XG5BTKat8DKVuG7SgmHOshI0xE3GiqCm0crNPvyhkRISe_dduPiKpo7b4=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Error estimate for the method<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/1xukXSqxhGfpM0Ah_3tiyJQFGxUgqULHq206zhOfAj3ueXKt02KJLK8krlHhGI6QIvTXgbIJxOkHIm_eJqytTuDsycXUJ1PD5DAT_1wVpLSqS9rJQpDe6L_Lsj52cLwy5956IY6n=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Example: Use Euler\u2019s procedure to&nbsp; find y(0.4) from the differential equation<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/x4TCffhS-gvy6x4DOiawGm_5N4y-WTW2Fhxn8qhgj2cD0XERiEDzVrjvOf1sIl2g9apnWGiEvXpOARlI7IW966J8BxovAfkGVa79jR791rg39KVBUJWiJjvZEi2acQ13IUcOBy6N=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Sol:<\/p>\n\n\n\n<p>Given equation&nbsp; dy\/dx = xy<\/p>\n\n\n\n<p>Here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/Ah7x1Gql3rRfAac1J36a1lMUu1dTmO9FDGKxEU-M3EZCBOq3I4wnGKzfknYCAh9gzjW_NSmzdEO_mrxaIJ_32fUwzQ8uLPIPGrZfH-QRru2yzAF2TehWBPUqIbo1WDE_j_2ll1Q_=s0\" width=\"286\" height=\"22\"><\/p>\n\n\n\n<p>&nbsp;We&nbsp; break&nbsp; the&nbsp; interval in four steps.<\/p>\n\n\n\n<p>&nbsp;So that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/GszovKCZQk7XW13pdsjBQabs0_sbX8YgKI9eYX24P-eZNZXKMmT42Q9MXNo7kIw1ghdfVwMd0rtIAZ-yLDiAszOrEx0JcwuoU8nVocwREKEPoWNe1KjngZHr4DyOgc5CZySMPp23=s0\" width=\"338\" height=\"22\"><\/p>\n\n\n\n<p>By Euler\u2019s formula<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/KXbU50TEQzuL-WO0nXLOZ-Vz7qw3MN8bvpiapuooK4fCEmi14tIA4kS_Y_4Zw0evcypHLD3Gdws2ARYDsjDVcbOvTI0mC-8Ru-EGvVQsPjT3gqJwc_ABYiTybPOV01lEWzQbBjB_=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>For n=0 in equation (i) we get, the first approximation<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/nIQZRyMpg8fItOzmYXhpClXfqVDeDA4udTlL8how5orCCrMy9feyCAs8g_Ibc9xTXl42nUtdbELVuXooJMYG4ppngFGT84f2w54Pm4DjaCBSzPPac9Kak1B9nugXFHTSIMN1FAbf=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>n=1 in equation (i) we obtain<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/lEoWhZ5gskyF7kbOPyOKyYb-PFx6SOi_EnEpN7wVGe8L_J-Eu0qkOaWGIMc2AUCJBsZ0bTVXA5X24wj-LaEy4E3aw2teHs1owZLiDRt23sl4TslhKh5e8pU8HXIXiMpIw0uh0yxW=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Put=2 in equation (i) we&nbsp; get, the third approximation<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/_9gwg4HgibA9X_jtlUI-iwWpG44qIjSlNIB-BWvonIU3y_HVFAG7qrzQZXi-gxZV2TXvTsVJABB3lKFXOulgFRJEZdgTWzWphdwQ2-Kxbs5hsjQDVP7J4hKjqj8O5c48cfCYLksk=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Put n=3 in equation (i) we&nbsp; get, the fourth approximation<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/xtJjCIAcIrbGXeChQ-Js9fmvj1GPI9viSDdUc58jHk00CWTnOHjlIQisFpwotTVQ_i5RmjtU8r6_cMYQGNqqUlRJcD4Bh6gjlXQtxk5PwSt4n60wIyIgYt0PCD5jXDvwyJfMejHh=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Hence y(0.4)&nbsp; =1.061106.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Modified Euler\u2019s&nbsp; Method:<\/strong><\/h2>\n\n\n\n<p>Instead of&nbsp; approximating f(x, y) by <img loading=\"lazy\" decoding=\"async\" width=\"63\" height=\"21\" src=\"https:\/\/lh3.googleusercontent.com\/c49cCMo72S5TwPVaYkiBSP8HyQ1A8dLCYNz236EbnGCsjT7i7b0h5mBARQl009PW0NyH1Ljv-wJ_GszwbBcFTWE4s8k38MID7Y9gyD3GSGuNtUSlh15mBOn7AobZgY3L7xbqt2E4=s0\"> as in Euler\u2019s method.&nbsp; In&nbsp; the&nbsp; <a href=\"https:\/\/atozmath.com\/example\/CONM\/RungeKutta.aspx?he=e&amp;q=meuler\" target=\"_blank\" rel=\"noreferrer noopener\">modified Euler\u2019s&nbsp; method<\/a> we&nbsp; have the iteration formula<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/UirPkDHKkQQ5cK0MQ95jmh7uoDruX0ExqPYyVwrKxF2VND1a32FFDUfeTwQwhRcHGSFYlrx-ZKB2vcAVJyIqxVJrwcewrgnNEQPncRo0uh31nic9YOjwPTBVxPO3MIuxFAJMvBq-=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/-khDqWhp07BwMDszKD-6MiR6Zzcsntnmd6mn9xHyhHILLgKutxpdNNJEqRPVJQa3dNK8QnOlHiQeqOIlqOv_QN_HgVcHJLpIl4Nqsj1ri5JIDhYWALQ8kGm30x4sP8d7A7UR5gdA=s0\" width=\"35\" height=\"28\"> is the nth approximation to y1 .The iteration started with the Euler\u2019s&nbsp; formula<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/B_zaZ_K5O-Dmd4NkQED988iUSwrhnAhFGxHJ4HIkKCFqyZIup51BQwzbQuelxUIOJECdDUovF-oA7ILxdLQYov2uNivqjQv0U_x73n7EJkUvywVMk385_s6kokCedlne8Bxn0VZ6=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Example: Use modified&nbsp; Euler\u2019s method&nbsp; to compute y&nbsp; for&nbsp; x=0.05.&nbsp; Given that<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/EU3DIkA2kqDt6ezg0pR4pBw2Whbzbtzjgg2Xj7qwpKzrcNpJy-tvGp3IeoJEgFwqlwo3VTDUCBuiI0irWYxYw53Vz-F6RMdbrxXdqrQBlrj-tFQlIrXOJCFu2ATORao8WLCZzxg3=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Result correct to three decimal places.<\/strong><\/p>\n\n\n\n<p><strong>Sol:<\/strong><\/p>\n\n\n\n<p>Given&nbsp; equation &nbsp; dy\/dx = x + y<\/p>\n\n\n\n<p>Here f(x, y) = x + y with y(0) = 1<\/p>\n\n\n\n<p>Take h = 0.05 &#8211; 0 = 0.05<\/p>\n\n\n\n<p>By modified Euler\u2019s&nbsp; formula the&nbsp; initial iteration is<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/09sEsd0wTwdLh1-Z_Lq8Kh8H3conKH6Fu3gGFK8VxrtHHtRAu1MVKRmVAQNUbSpDULEGVjPL8T1vEYej2I3gg3IkR4t-5cmxq2bWbdT_SNXz0sx3I2V0Cz6rBzpFtJmlkaAqqxyx=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>The iteration&nbsp; formula by&nbsp; modified Euler\u2019s method&nbsp; is<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/TuiQHVwT6n3ZaXEgLzNOEqHGDuwje4eWxeDeJ-lYECrTO_elq8sFsUIyCIIwsWzkVe92Qw6rZAUEXiJUP91CFMOt1J9Qbe9SNdVKDfQuAKQ7UDarZE7vizr5InnpvR8-xrr06L4q=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>For n=0&nbsp; in equation&nbsp; (i)&nbsp; we get<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/lQy54ZKrJp_HJuKFTH_bF-9nVIuauglZKoXMQpT4L1DrWHnaqBjQDMtszvM52aq0bGh4-F8vTZnRtu70eRrND-qwQCm2oSBkkiFCdYZ1yq7x3gXtBddclUmdS86wFxEsLb2UYBdr=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/odW107OdkRcJxw0evMDmKt6NBx9qR27w8URP7yYOntE2ZoYpK3z8s6vyRclj45hJASanUkE6tXVm9wurTbxv00Cnl6ZjIvbSitVJ1IFVkonxZgWt_PmF_Uba0l9z8B1fVhzUbS4s=s0\" width=\"340\" height=\"25\">&nbsp; as&nbsp; above<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/zBRb-SZHjSWYM-Q9jyba_B5ftDOttoJI7mV9AjaVtp_pMLG4sdCZXYofz31qaSpq6i8RJEXbPSimofLNkBANK9hz-f6VozFT42dyd8CvtN4OoEnMFldYOqhNJM5CQ-jPMxVtnR0N=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>For n=1&nbsp; in equation&nbsp; (i)&nbsp; we get<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/J6kqgrQGkFXqgS-LXFuXLk2oMzpqLx0WEnd5lAgGBNhseTthQKviYGx9KWFq3gknuiNNyc9DVVWD3W0Jw6NxbWELSpE4eMQ-ThZGwmFAwkIOBveQSNzZ3zp_J9DUNWsVjj7_35Fq=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>&nbsp;For n=3&nbsp; in equation&nbsp; (i)&nbsp; we get<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/pseqb6xS5NLxNN76NUMmVJEtgxOI0XHVUsBRJ0FpLmipPTlmTu2wRFTnFLx8YiEfnlsLC5BXj_xz43GgqppAWOcsAdRj-trMraLrpKKd6cXhsYR5cIsB18o462IE1YigI3ZQi54O=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Since&nbsp; third and&nbsp; fourth approximation are equal .<\/p>\n\n\n\n<p>Hence y=1.0526&nbsp; at x =&nbsp; 0.05&nbsp; correct&nbsp; to&nbsp; three decimal places.<\/p>\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\">\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-are-simpsons-rules-for-numerical-integration\/\" target=\"_blank\" rel=\"noreferrer noopener\">What are simpson&#8217;s rules?<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-partial-differentiation\/\" target=\"_blank\" rel=\"noreferrer noopener\">Partial differentiation<\/a><\/li>\n<\/ul>\n\n\n\n<p>Study with Goseeko: <a href=\"https:\/\/www.goseeko.com\/landing\/international-students\/\">https:\/\/www.goseeko.com\/landing\/international-students\/<\/a><br>UAE Students Register here: <a href=\"https:\/\/www.goseeko.com\/landing\/middle-east\/\">https:\/\/www.goseeko.com\/landing\/middle-east\/<\/a><\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This method was given by Leonhard Euler. Euler\u2019s method is the first order numerical method for solving ordinary differential equations with given initial value.<\/p>\n","protected":false},"author":3,"featured_media":5513,"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-5127","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 Euler\u2019s modified method? - Goseeko blog<\/title>\n<meta name=\"description\" content=\"This method was given by Leonhard Euler. 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