{"id":5729,"date":"2021-09-29T18:39:37","date_gmt":"2021-09-29T13:09:37","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5729"},"modified":"2021-11-21T20:11:00","modified_gmt":"2021-11-21T14:41:00","slug":"differential-equations-reducible-to-variable-separable-form","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/differential-equations-reducible-to-variable-separable-form\/","title":{"rendered":"Differential Equations Reducible to Variable Separable Form"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

Differential Equations Reducible to Variable Separable Form<\/h2>\n

Differential Equations of the form \\(dy\\over dx\\) = f(ax + by + c) can be reduce to variable separable form by the substitution ax + by + c = 0 which can be cleared by the examples given below.<\/p>\n

Example<\/strong><\/span> : Solve the differential equation : \\(sin^{-1}\\) \\(dy\\over dx\\) = x + y<\/p>\n

Solution<\/span><\/strong> : We are given that,<\/p>\n

\\(sin^{-1}\\) \\(dy\\over dx\\) = x + y<\/p>\n

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

Let x + y = v. Then<\/p>\n

1 + \\(dy\\over dx\\) = \\(dv\\over dx\\)<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = \\(dv\\over dx\\) – 1<\/p>\n

Putting x + y = v and \\(dy\\over dx\\) = \\(dv\\over dx\\) – 1 in the given differential equation, we get<\/p>\n

\\(dv\\over dx\\) – 1 = sin v<\/p>\n

\\(dv\\over dx\\) = 1 + sin v<\/p>\n

\\(1\\over 1 + sin v\\) dv = dx<\/p>\n

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

\\(\\int\\) dx = \\(\\int\\) \\(1\\over 1 + sin v\\) dv <\/p>\n

\\(\\int\\) dx = \\(\\int\\) \\(1 – sin v\\over 1 – sin^2 v\\) dv <\/p>\n

\\(\\int\\) dx = \\(\\int\\) \\(1 – sin v\\over cos^2 v\\) dv <\/p>\n

\\(\\int\\) dx = \\(\\int\\) \\((sec^2 v – tan v sec v)\\) dv<\/p>\n

x = tan v – sec v + C<\/p>\n

\\(\\implies\\) x = tan (x + y) – sec (x + y) + C, which is the required solution.<\/p>\n

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

Solution<\/span><\/strong> : We are given that,<\/p>\n

\\(dy\\over dx\\) = cos(x + y)<\/p>\n

Let x + y = v. Then<\/p>\n

1 + \\(dy\\over dx\\) = \\(dv\\over dx\\)<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = \\(dv\\over dx\\) – 1<\/p>\n

Putting x + y = v and \\(dy\\over dx\\) = \\(dv\\over dx\\) – 1 in the given differential equation, we get<\/p>\n

\\(dv\\over dx\\) – 1 = cos v<\/p>\n

\\(dv\\over dx\\) = 1 + cos v<\/p>\n

\\(1\\over 1 + cos v\\) dv = dx<\/p>\n

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

\\(\\int\\) dx = \\(\\int\\) \\(1\\over 1 + cos v\\) dv <\/p>\n

\\(\\int\\) dx = \\(\\int\\) \\(1\\over 2\\) \\(sec^2 v\/2\\) dv <\/p>\n

x = \\(tan v\/2\\) + C<\/p>\n

\\(\\implies\\) x = x = \\(tan (x + y)\/2\\) + C, which is the required solution<\/p>\n\n\n

\n
Next – Solution of Homogeneous Differential Equation<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Differential Equations in Variable Separable Form<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find the solution of the differential equations reducible to variable separable form with examples. Let’s begin – Differential Equations Reducible to Variable Separable Form Differential Equations of the form \\(dy\\over dx\\) = f(ax + by + c) can be reduce to variable separable form by the substitution ax + …<\/p>\n

Differential Equations Reducible to 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":[251,257,256,258],"yoast_head":"\nDifferential Equations Reducible to 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 reducible to 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-reducible-to-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 Reducible to Variable Separable Form - 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