{"id":5723,"date":"2021-09-29T18:38:49","date_gmt":"2021-09-29T13:08:49","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5723"},"modified":"2021-11-21T20:12:14","modified_gmt":"2021-11-21T14:42:14","slug":"differential-equations-of-form-dy-dx-fx-or-fy","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/differential-equations-of-form-dy-dx-fx-or-fy\/","title":{"rendered":"Differential Equations of Form dy\/dx = f(x) or f(y)"},"content":{"rendered":"

Here you will learn how to find the general solution of differential equations of form dy\/dx = f(x) or f(y) with examples.<\/p>\n

Let’s begin –<\/p>\n

Differential Equations of Form dy\/dx = f(x) or f(y)<\/h2>\n

(1) Differential Equations of Form dy\/dx = f(x)<\/strong><\/h4>\n

To solve this type of differential equations we integrate both sides to obtain the general solution as discussed below.<\/p>\n

\n

We have. <\/p>\n

\\(dy\\over dx\\) = f(x) \\(\\iff\\) dy = f(x)dx<\/p>\n

Integrating both sides, we obtain<\/p>\n

\\(\\int \\) dy = \\(\\int \\) f(x) dx + C or,<\/p>\n

y = \\(\\int \\) f(x) dx + C, <\/p>\n

which gives general solution of the differential equation.<\/p>\n<\/blockquote>\n

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

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

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

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

Integrating both sides, we get<\/p>\n

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

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

\\(\\implies\\) y = \\(1\\over 2\\) \\(log|x^2 + 1|\\) + C<\/p>\n

Clearly, y = \\(1\\over 2\\) \\(log|x^2 + 1|\\) + C is defined for all x \\(\\in\\) R.<\/p>\n

Hence, y = \\(1\\over 2\\) \\(log|x^2 + 1|\\) + C, x \\(\\in\\) R is the solution of the given differential equation.<\/p>\n

(2) Differential Equations of Form dy\/dx = f(y)<\/strong><\/h4>\n

To solve this type of differential equations we integrate both sides to obtain the general solution as discussed below.<\/p>\n

\n

We have. <\/p>\n

\\(dy\\over dx\\) = f(y) <\/p>\n

\\(\\implies\\) \\(dx\\over dy\\) = \\(1\\over f(y)\\), provided that f(y) \\(\\ne\\) 0<\/p>\n

\\(\\implies\\) dx = \\(1\\over f(y)\\) dy<\/p>\n

Integrating both sides, we obtain<\/p>\n

\\(\\int \\) dx = \\(\\int \\) \\(1\\over f(y)\\) dy + C or,<\/p>\n

x = \\(\\int \\) \\(1\\over f(y)\\) dy + C, <\/p>\n

which gives general solution of the differential equation.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Solve the given differential equation :  \\(dy\\over dx\\) = \\(1\\over y^2 + sin y\\)<\/p>\n

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

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

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

\\(\\implies\\) dx = \\(y^2 + sin y\\)dx<\/p>\n

Integrating both sides, we get<\/p>\n

\\(\\int \\) dx = \\(\\int \\) \\(y^2 + sin y\\)dx<\/p>\n

\\(\\implies\\) x = \\(y^3\\over 3\\) – cosy + C<\/p>\n

Hence, x = \\(y^3\\over 3\\) – cosy + C is the solution of the given differential equation.<\/p>\n\n\n

\n
Next – Differential Equations in Variable Separable Form<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – First Order First Degree Differential Equations<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find the general solution of differential equations of form dy\/dx = f(x) or f(y) with examples. Let’s begin – Differential Equations of Form dy\/dx = f(x) or f(y) (1) Differential Equations of Form dy\/dx = f(x) To solve this type of differential equations we integrate both sides to obtain …<\/p>\n

Differential Equations of Form dy\/dx = f(x) or f(y)<\/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],"yoast_head":"\nDifferential Equations of Form dy\/dx = f(x) or f(y) - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find the general solution of differential equations of form dy\/dx = f(x) and dy\/dx = f(y) 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-of-form-dy-dx-fx-or-fy\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Differential Equations of Form dy\/dx = f(x) or f(y) - 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