{"id":5733,"date":"2021-09-29T18:47:37","date_gmt":"2021-09-29T13:17:37","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5733"},"modified":"2021-11-21T20:06:39","modified_gmt":"2021-11-21T14:36:39","slug":"solution-of-linear-differential-equation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/solution-of-linear-differential-equation\/","title":{"rendered":"Solution of Linear Differential Equation"},"content":{"rendered":"

Here you will learn how to find solution of linear differential equation of first order first degree with examples.<\/p>\n

Let’s begin –<\/p>\n

Solution of Linear Differential Equation<\/h2>\n

(1) Linear Differential Equation of the form \\(dy\\over dx\\) + Py = Q<\/strong><\/h4>\n

A differential equation is linear if the dependent variable (y) and its derivative appear only in first degree.<\/p>\n

\n

The general form of a linear differential equation is <\/p>\n

\\(dy\\over dx\\)  + Py = Q<\/p>\n

where P and Q are functions of x (or constants)<\/p>\n<\/blockquote>\n

This type of differential equations are solved when they are multiplied a factor, which is called integrating factor, because by multiplication of this factor the left hand side of the differential equation (i) becomes exact differential of some function.<\/p>\n

Algorithm :<\/strong><\/p>\n

\n

1). Write the differential equation in the form \\(dy\\over dx\\)  + Py = Q and obtain P and Q<\/p>\n

2). find the integrating factor (I. f.) given by I.f = \\(e^{\\int Pdx}\\)<\/p>\n

3). Multiply both sides of equation in step 1 by I.f.<\/p>\n

4). Integrate both sides of the equation obtained in step 3 with respect to x to obtain <\/p>\n

y(I.f) = \\(\\int\\) Q(I.f) dx + C, which gives the required solution.<\/p>\n<\/blockquote>\n

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

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

\\(dy\\over dx\\) – \\(y\\over x\\) = \\(2x^2\\)<\/p>\n

Clearly it is a differential equation of the form<\/p>\n

\\(dy\\over dx\\) + Py = Q , where P = \\(-1\\over x\\) and Q = \\(2x^2\\)<\/p>\n

Now, I.f = \\(e^{\\int Pdx}\\) = \\(e^{\\int (-1\/x)dx}\\) = \\(e^{-log x}\\) = \\(1\\over x\\)<\/p>\n

By algorithm, the solution is<\/p>\n

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

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

\\(\\implies\\) y = \\(x^3\\) + Cx, which is the required solution.<\/p>\n

(2) Linear Differential Equation of the form \\(dx\\over dy\\) + Rx = S<\/strong><\/h4>\n

Sometimes a linear differential equation can be put in the form \\(dx\\over dy\\) + Rx = S where R and S are functions of y or constants<\/p>\n

Note that here y is independent variable and x  is a dependent variable.<\/p>\n

Algorithm :<\/strong><\/p>\n

\n

1). Write the differential equation in the form \\(dx\\over dy\\) + Rx = S and obtain R and S<\/p>\n

2). find the integrating factor (I. f.) given by I.f = \\(e^{\\int Rdy}\\)<\/p>\n

3). Multiply both sides of equation in step 1 by I.f.<\/p>\n

4). Integrate both sides of the equation obtained in step 3 with respect to x to obtain <\/p>\n

x(I.f) = \\(\\int\\) S(I.f) dy + C, which gives the required solution.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Solve the differential equation : ydx + \\(x – y^3\\) dy = 0<\/p>\n

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

ydx + \\(x – y^3\\) dy = 0<\/p>\n

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

Clearly it is a differential equation of the form<\/p>\n

\\(dx\\over dy\\) + Ry = S , where R = \\(1\\over y\\) and S = \\(y^2\\)<\/p>\n

Now, I.f = \\(e^{\\int Rdy}\\) = \\(e^{\\int (1\/y)dy}\\) = \\(e^{log y}\\) = y<\/p>\n

By algorithm, the solution is<\/p>\n

xy = \\(\\int\\) \\(y^3\\) dy + C<\/p>\n

\\(\\implies\\) xy = \\(y^4\\over 4\\) + C, which is the required solution.<\/p>\n\n\n

\n
Previous – Solution of Homogeneous Differential Equation<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find solution of linear differential equation of first order first degree with examples. Let’s begin – Solution of Linear Differential Equation (1) Linear Differential Equation of the form \\(dy\\over dx\\) + Py = Q A differential equation is linear if the dependent variable (y) and its derivative appear only …<\/p>\n

Solution of Linear Differential Equation<\/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":[251,250,252,249],"yoast_head":"\nSolution of Linear Differential Equation - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find solution of linear differential equation of first order first degree 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\/solution-of-linear-differential-equation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Solution of Linear Differential Equation - 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