{"id":5727,"date":"2021-09-29T18:43:29","date_gmt":"2021-09-29T13:13:29","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5727"},"modified":"2021-11-21T20:11:28","modified_gmt":"2021-11-21T14:41:28","slug":"differential-equations-in-variable-separable-form","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/differential-equations-in-variable-separable-form\/","title":{"rendered":"Differential Equations in Variable Separable Form"},"content":{"rendered":"

Here you will learn how to find the solution of the differential equations in variable separable form with examples.<\/p>\n

Let’s begin –<\/p>\n

Differential Equations in Variable Separable Form<\/h2>\n

If the differential equation can be put in the form f(x) dx = g(y) dy, we say that the variables are seperable and such equations can be solved by integrating on both sides. The solution is given by <\/p>\n

\n

\\(\\int\\) f(x) dx = \\(\\int\\) g(y) dy + C, where C is an arbitrary constant<\/p>\n<\/blockquote>\n

Note<\/strong> : There is no need of introducing arbitrary constants of integration on both sides as they can be combined together to give just one arbitrary constant.<\/p>\n

Example<\/strong><\/span> : Solve the differential equation : (x + 1)\\(dy\\over dx\\) = 2xy<\/p>\n

Solution<\/span><\/strong> : We have, <\/p>\n

(x + 1)\\(dy\\over dx\\) = 2xy<\/p>\n

\\(\\implies\\) (x + 1)dy = 2xy dx<\/p>\n

\\(\\implies\\) \\(dy\\over y\\) = \\(2x\\over x + 1\\) dx<\/p>\n

Now, integrating on both sides,<\/p>\n

\\(\\int\\) \\(1\\over y\\) dy = 2 \\(\\int\\) \\(x\\over x + 1\\) dx<\/p>\n

\\(\\implies\\) \\(1\\over y\\) dy = 2 \\(\\int\\) \\(x + 1 – 1\\over x + 1\\) dx<\/p>\n

\\(\\implies\\) \\(1\\over y\\) dy = 2 \\(\\int\\) \\(1 – {1\\over x + 1}\\) dx<\/p>\n

log y = 2{x – log| x + 1 |} + C, which is the solution of the given differential equation.<\/p>\n

Example<\/strong><\/span> : Solve the differential equation : cos x(1 + cos y) dx  – sin y(1 + sin x)dy = 0<\/p>\n

Solution<\/span><\/strong> : We have, <\/p>\n

cos x(1 + cos y) dx  – sin y(1 + sin x)dy = 0<\/p>\n

\\(\\implies\\) \\(cos x\\over 1 + sin x\\) dx – \\(sin y\\over 1 + cos y\\) dy = 0<\/p>\n

\\(\\implies\\) \\(cos x\\over 1 + sin x\\) dx + \\(-sin y\\over 1 + cos y\\) dy = 0<\/p>\n

Now, integrating on both sides,<\/p>\n

\\(\\int\\) \\(cos x\\over 1 + sin x\\) dx + \\(\\int\\) \\(-sin y\\over 1 + cos y\\) dy = 0<\/p>\n

\\(\\implies\\) log|1 + sin x| + log|1 + cos y| = log C<\/p>\n

\\(\\implies\\) log{|1 + sin x|.|1 + cos y|} = log C<\/p>\n

|1 + sin x|.|1 + cos y| = C<\/p>\n

Hence, (1 + sin x)(1 + cos y) = C is the solution of the given differential equation.<\/p>\n\n\n

\n
Next – Differential Equations Reducible to Variable Separable Form<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Differential Equations of Form dy\/dx = f(x) or f(y)<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find the solution of the differential equations in variable separable form with examples. Let’s begin – Differential Equations in Variable Separable Form If the differential equation can be put in the form f(x) dx = g(y) dy, we say that the variables are seperable and such equations can be …<\/p>\n

Differential Equations in Variable Separable Form<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[38],"tags":[251,255,256,258],"yoast_head":"\nDifferential Equations in Variable Separable Form - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find the solution of the differential equations in variable separable form with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/differential-equations-in-variable-separable-form\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Differential Equations in Variable Separable Form - 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