{"id":6545,"date":"2021-09-30T13:20:45","date_gmt":"2021-09-30T07:50:45","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5845"},"modified":"2026-03-06T04:09:04","modified_gmt":"2026-03-05T22:39:04","slug":"what-is-an-exact-differential-equation","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-an-exact-differential-equation\/","title":{"rendered":"What is an exact differential equation?"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Overview<\/strong><\/h2>\n\n\n\n<p>An <a rel=\"noreferrer noopener\" href=\"https:\/\/tutorial.math.lamar.edu\/classes\/de\/exact.aspx\" target=\"_blank\">exact differential equation<\/a> is formed by differentiating its solution directly without any other process,<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/2k5Ugy7LirYnK7zyIHep3o5mVf_l1WEvj7foQst0K2ZdR97DuOJmWntZ7wmiEq2n30FcajLMCMnYA-AG3x9MPm-TnXB4gOmSPqKh2qDVPiEfLqC3dxElaPWH8Sml339Fux8M0xas=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Is called an <a href=\"https:\/\/tutorial.math.lamar.edu\/classes\/de\/exact.aspx\" target=\"_blank\" rel=\"noreferrer noopener\">exact differential equation<\/a> if it satisfies the following condition-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/Ry975u9SjadM-JnEfUwXB7Z6axYJ2wglus79m2tefOcgz-4FvvJM1xUQ8KAXL73lmB0CnEBud59-DCyQmqAUn6bPX3kaARkG2Kn7Eq5d_twv_PVCpziGWYai9JLrmHt6002rC99B=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Here<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/R9gadLiL9pumu9s5UKtkpPOOUHJov0XuecaWNC9iDy1cJU6qvPXz-9efa631eAHVZn61cB-FcNdt_VHiIS6T93fNOAli4HijKFeEZIkgIq1ZS2zkrGBmmo6-gNi-h4u_fHp-nk-g=s0\" width=\"22\" height=\"34\"> is the differential co-efficient of M with respect to y keeping x constant and<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/T6touYJVDLHPAtFhSkTvZvGMHh8QIl8H_spoB43lZRZy73bnUhEii_MluqQfWT7-b8S9D4WPgJRs9wn4rz_S_17hPrKDZ7GXd0rOp7A1LvCA13CaI1-1CeRwHdgbhmpLUiGM1B84=s0\" width=\"22\" height=\"32\"> is the differential co-efficient of N with respect to x keeping y constant.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Step by step method to solve an exact differential equation<\/strong><\/h2>\n\n\n\n<p>1. Integrate M w.r.t. x keeping y constant.<\/p>\n\n\n\n<p>2. Integrate with respect to y, those terms of N which do not contain x.<\/p>\n\n\n\n<p>3. Add the above two results as below-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/c0BHpzFjlN8jr7GWgvFMZaQeMyUcLVT_oRnzMO5pl9FKqkfyVvcIKkg4icWvdi7CLZOHr7TI3zuUv3EjfxjJ4vfVOb3OG_BbJXCL_1KOVN3xZhjs4XKlfiSc0m_G75FRDJ74aKtk=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Provided&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/pDRahJ_1koawMVkzbkdJnKPxy52hUei0s5C2CGW93ggGAUDmjUk46aCTxkjtho8UGw1ReBOd4MfeFhD4TYTuiXqxfsNN9sNPbbdhLF984h4TZjPEbYZQdLFcjk3QqJKrirUemruw=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Example-3: Determine whether the differential function ydx \u2013xdy = 0 is exact or not.<\/strong><\/p>\n\n\n\n<p>Solution.&nbsp; Here the equation is the form of&nbsp; M(x , y)dx + N(x , y)dy = 0<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp; But, we will check for exactness,<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/zUZd2pLNCQ60iFpNZ-rvbo40wVUTgpkLp89mAFS7Wb9ziQ-Q3ADPKwVBIq7WFkj0Kum81O7IdpJ3qnBFxLGBZKUyXiYRFhtbdliD1LAHolEf88QTpz7OBS5J54fSuURAZq71P-hJ=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>These are not equal results, so we can say that the given diff. eq. is not exact.<\/p>\n\n\n\n<p><strong>Example: Solve-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/hB7vI10OlaOGiZFwis898q7lEJxe_U0pUj3eoeVjQObe9p9j_XMnFI4iPj-5GS_F6LU4PaSR5KAlwrIfPdt-vk62TUaUPUHnjB1EbZqD7RnafAHE_ROe81qBT1cph3_nWRHoPYhw=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Sol.<\/p>\n\n\n\n<p>Here, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/c_sFmRYkk-o48KM4nrxgXLi89I_mTXAdI0RD1geUVTzq7700fuOlqUh2wehaZH8jdUE8D4157WGc0Wm3XDQjQmcoq96fa6kuk9vJHyhov0v3FsUgYaq9s7XTzz6clTbyG_8xrLcl=s0\" width=\"300\" height=\"27\"><\/p>\n\n\n\n<p>So that-&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/NdH-wHs5jKcvzUhVPzq-YsdsROGWZ0FPCstdyakRD_tUYlVk3EH29-A9qZ1uimTUTuIjf53a2JHkJP-1fERGgUgPQsWVNMdMZfXwch-hAe8oCG7yQwCDnB1XBcf3KQPK2Gnkx71n=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Thus the equation is exact and its solution is-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/21lH7Mt1VzzBelbvkks1rvR1akvkKCtyjMcynmyJ_VW3zPi6LP6mHdwCLdwKigFCeOhd8JCJVU4bOVtn9Z_rRfWmnRvJT6R6wJn_bewKG9puebEOchKhXrWPjODiWNcsYHNPAMbM=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Which means-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/4kGuVmftPTLr68niRzNRARmlFerVj76nQwvR11Xy6pNh8B1L_twVlezHIWvlr9BviZKPmlTnpgZ0qfjqctMslLpk0gBWa9DMdYrthogrONVyEmirwPBj2i4spW0twK6HWGJYESMd=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Or&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/JWDxIZ21gtfVglTQkS33rJuNNf-vMSiD_3tbZo7AsY-GSynYhCNe2dVDDpLdJpuQNCgnQVSjVY_hT-It7fO-7gUX3tOnyPNcoJcWjeS-fi5Vb46CXzKsXo0lfrjS00-vijm-JnJC=s0\" alt=\"\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong><a rel=\"noreferrer noopener\" href=\"https:\/\/nptel.ac.in\/content\/storage2\/courses\/122104018\/node62.html\" target=\"_blank\">Equation reducible to the exact form<\/a><\/strong><\/h2>\n\n\n\n<p><strong>1. If M dx + N dy = 0 be an homogenous equation in x and y, then 1\/ (Mx + Ny) is an integrating factor (M dx + N dy&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Example: Solve-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/zKE-2cCEmypHXwC0uay3gZhhAhihmTRzxF3JBlvzS16LmOp6OSqk9y0AD0iOhdGvtloL__PFXnYMgOicpDhjMzWTfmZHTOzpRWwqf07ZUbkjh2gL_xTK6_6JSnJbZ67Mi47WCbuv=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Sol.<\/strong><\/p>\n\n\n\n<p>We can write the given equation as-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/oJzqGboHXv6l1oZZqaKVguoMQXDwCfKMk9IeHpKkfxH29F1ATdxtUB1k30OzSghU0f32CUg8GRF5dpBp5PqBFNCjVzmbKRJP91LZJcx9diSEV4Ez2EvcAH-2mIoV4ra-i1yIWL88=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Here,<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/lMNBtOSN__caOlvymrSE0pNE7vjCcQOdCGd3_lzkmaif39SO_E70MIJfCgmMDgCfNZMb9JsWCaa-uwsz3cvrUh7iGN8XTleCnvKNYMUXfGa9weRIfiK-1DQfwtDfM8SyoXmxc2CA=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>Multiply equation (1) by 1\/x^4 we get-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/rEZXG09Gd_A4NTN_HHob_fhvQwsmQTiZrTTwbW_JWE5d1qtH8v7ruYSoq-_MOLUMlkjRHn-KxjirZtfUM3JvzKOe3SprhudtUcFAupXyJGPoPasDmaaKBYzyp7CgqPF3EjuYMtpr=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p>This is an exact differential equation-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/UBDiM2gYoMJ4pHuk2ZXs2qXRrgPlvCqVU2U3YdF9a2nfCxXJqlhPWvJq6VaNosjynt85RBZHdD9C7tUf5EaxkqgUBpCsvnVx-e2543-WR4pEKvovFV9kMz8KCW-pliSzZQ988t0n=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>2. I.F. for an equation of the type <\/strong><strong><\/strong><\/p>\n\n\n\n<p><strong>IF the equation Mdx + Ndy = 0 be this form, then 1\/(Mx \u2013 Ny) is an integrating factor.<\/strong><\/p>\n\n\n\n<p><strong>&nbsp;Example: Solve-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/wYPaGaRnkCxcEb_H_dKap7e3ohY_FM9DvzPH-sqpkzKU6YmEHe2k2inBY26aNbORQ_5pdpAuFPnMzY1qow2cz3sqA3ILmTzooRT6Zb6WnGF-o_QhD-jY8Qj3z10xGgbq2cy0tJL1=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Sol.<\/strong><\/p>\n\n\n\n<p><strong>Here we have-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/nqbaKyRDKzAG3a_szndP7ATt5uBLQLPiuj3z0tuukurc4-oadmbSZKFcgSGd2ZlpTuvEpwQyQh83lUYeY5bTwAm5wGUOLk-FAxrjIUsYKUBxv_a_nDBnMujtX95Iyyh_MrY9AzzU=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Now divide by xy, we get-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/NTS7ZSn1_PPc1B8m4LcH_HdcQgVyAD9DTeNsWb6LUcdIwCM0tvUSLTmmkhV7SDk5pCK53soNJ85Z1DSPC6_jiobsLcz43bpjAwP2OfjYeG-YSF0LZLMVdUSAjCvj4QH-grj02bsA=s0\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/h_aHYhb5DUHHuKGe1UgS-vOWE9YbCdtwaaybQzv4OLKmenADVsdjeamCMdMoidaEo3Vc2PV-fv77qV7I5OUt2kJGnMYttGDKojffy6A9KPU7t0uZXBcgo20PVppClURWHfOA-9aW=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Which is an exact differential equation-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/e1ZvBrNeTxhA0Z4P8CVNIp0F_fYnFqEGtzCW-q4e-d7RZ3EgdSPv8dAL82Xa_7C1A6V4lOCFu5i6rq1isMogKoEsTe4FENv0lEAzMYWj2Bg6AkPpINEbo3ijr_Tdcskc2REg8n6N=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>3. In the equation M dx + N dy = 0,<\/strong><\/p>\n\n\n\n<p><strong>(i) If&nbsp; <\/strong><strong><\/strong><strong>be a function of x only = f(x), then<\/strong><strong><\/strong><strong>&nbsp; is an integrating factor.<\/strong><\/p>\n\n\n\n<p><strong>(ii) If be a function of y only = F(x), then is an integrating factor.<\/strong><\/p>\n\n\n\n<p><strong>Example: Solve-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/rqfTsMqWEhxZIATdaLYVKa-HkI8Sd4qqvjxWNL8eeSHRGDEXZ7J6z_rlZD1C1LN7BtgoaZh4oWEusjqUPs9x0YVLvey0ejgiXjuSm9KSC7d96-JNkrQHeGlcf8SMTLNOWdLjuLhd=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Sol.<\/strong><\/p>\n\n\n\n<p><strong>Here given,<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/Ex3qHrLUPgjh9LID6N3JMTXjakKtAYwHm-9fo4Gw7uVzBjSuP3_kRh8VjhkZjPtjXsa5moKbTuaRLgkL2DbbdY_cx-UPIK520W-6kT8KFj5nggdlN5G5I8huKf3uOiHLfSBLC2Aw=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>M = 2y and N = 2x log x &#8211; xy<\/strong><\/p>\n\n\n\n<p><strong>Then-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/1PSjeBm_m9zSUMcd8pH54LLb12NsEJRejB8WN8Fg-ExrMZjtvACKceabjWJse3mrsTxZQrkHInXhjYQDOnBqQMXaoNOhKxDog-YpL8F3HCXV_2jY6gqk44k94S1gwT3GsO_nJSXA=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Here<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/D7a5Y7CtiqtC2SqNnlc-USWwb1FamF3z3e_aGkgE2KiITZshKvIZIrK-GymKLXIhlDmMV-W2MaIT3iP-5FRWAaniULpt1V0b8Jqtent-NR_Cip3jC-J3xKMCROOKK6_q7mHoZZLk=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Then<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/INtgVirWHkB5qUzM3MJoIhYY-mY66heibelpMQhcDlDz_JWNZ8CZy_o5nO0UPexw1KNA7HDZD0GlbD5CUWuwRKcTrQLT_VJVM5aBIQJtFiBJZqPsCFfVBeoP5aivvYESINxboYd5=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Now multiplying equation (1) by 1\/x, we get-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/8h9I1FLCggrd_0bAwkMOpQt2UAxtuuCvmmhURG1njyZ-RKrKuDU2OfeRVs0KbwGg101UeuuEvH--jHw6Q7Qxtr7RPqMzpk0UJPuAaGhx8lSFSLKbxmPVmibuqV1kLlkIkjl-7hMf=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>4. For the following type of equation-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/xiz3R0v8aB2Me6TITr6qveezOqSFyXQvKlCvpX_Co_De9IsCaitackSA6Xl33Wm6wW6l-dGjSbKO1togc8VPq1nOTyeYKe7peAQHW0JLIn7NRU2BE82TNHA3M8nKh8dNdiQ-K1AX=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>An I.F. is <\/strong><strong><\/strong><\/p>\n\n\n\n<p><strong>Where-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/fxzDynVFDjk7ErPMLGU9VFCtCunhO6EYIbyjBFXMLP_pwh6CcIrPkAGO5FjDy2PC8Z51ciJyMYMgQGlQnHfvhVTMUrTujhPAZGwdXzNGCRhT0KLRtgsd2j7dtf4tied4URYr5R0E=s0\" alt=\"\"\/><\/figure>\n\n\n\n<p id=\"block-74b9767b-165d-446c-8ff5-3652f80c7fc6\"><strong>Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/strong><\/p>\n\n\n\n<ul id=\"block-043fd18d-a1fd-47bb-b6ac-4d976259d566\" class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-a-linear-differential-equation\/\" target=\"_blank\" rel=\"noreferrer noopener\">What are linear differential equations?<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-partial-differentiation\/\" target=\"_blank\" rel=\"noreferrer noopener\">What is partial differentiation?<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-curvature-and-radius-of-curvature\/\" target=\"_blank\" rel=\"noreferrer noopener\">curvature<\/a>. <a href=\"https:\/\/slot88ku.news\/\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/slot88ku.news\/<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>An exact differential equation is formed by differentiating its solution directly without any other process,Is called an exact differential equation if it satisfies the following condition-<\/p>\n","protected":false},"author":28,"featured_media":4798,"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-6545","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 an exact differential equation? - Goseeko blog<\/title>\n<meta name=\"description\" content=\"An exact differential equation is formed by differentiating its solution directly without any other process,Is called an exact differential equation if it satisfies the following condition-\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link 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