{"id":5702,"date":"2021-09-29T18:37:05","date_gmt":"2021-09-29T13:07:05","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5702"},"modified":"2021-11-21T20:15:06","modified_gmt":"2021-11-21T14:45:06","slug":"solution-of-differential-equation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/solution-of-differential-equation\/","title":{"rendered":"Solution of Differential Equation"},"content":{"rendered":"

Here you will learn how to find solution of differential equation i.e. general solution and particular solution with examples.<\/p>\n

Let’s begin –<\/p>\n

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

The solution of differential equation is a relation between the variables involved which satisfies the differential equation.<\/p>\n

for example<\/strong>, y = \\(e^x\\) is a solution of the differential equations \\(dy\\over dx\\) = y.<\/p>\n<\/blockquote>\n

General Solution<\/strong><\/h4>\n
\n

The solution which contains as many as arbitrary constants as the order of the differential equations is called the general solution.<\/p>\n

for example<\/strong>, y = A cos x + B sin x is the general solution of the equation \\(d^2y\\over dx^2\\) + y = 0.<\/p>\n<\/blockquote>\n

Particular Solution<\/strong><\/h4>\n
\n

The solution obtained by giving particular values to the arbitrary constants in the general solution of a differential equations is called a particular solution.<\/p>\n

for example<\/strong>, y = 3 cos x + 2 sin x is the particular solution of the equation \\(d^2y\\over dx^2\\) + y = 0.<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Show that y = Ax\u00a0 + \\(B\\over x\\), x \\(\\ne\\) 0 is a solution of the differential equation \\(x^2\\)\\(d^2y\\over dx^2\\) + x\\(dy\\over dx\\) – y = 0<\/p>\n

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

y = Ax\u00a0 + \\(B\\over x\\) = 0, x \\(\\ne\\) 0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 ……..(i)<\/p>\n

Differentiating both sides with respect to x, we get<\/p>\n

\\(dy\\over dx\\) = A – \\(B\\over x^2\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0……….(ii)<\/p>\n

Again differentiating with respect to x, we get<\/p>\n

\\(d^2y\\over dx^2\\) = \\(2B\\over x^3\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 …………..(iii)<\/p>\n

Substituting the values of y, \\(dy\\over dx\\) and \\(d^2y\\over dx^2\\) in \\(x^2\\)\\(d^2y\\over dx^2\\) + x\\(dy\\over dx\\) – y , we get<\/p>\n

\\(x^2\\)\\(d^2y\\over dx^2\\) + x\\(dy\\over dx\\) – y = \\(x^2\\)(\\(2B\\over x^3\\)) + x(A – \\(B\\over x^2\\)) – (Ax + \\(B\\over x\\))\u00a0<\/p>\n

= \\(2B\\over x\\) + Ax – \\(B\\over x\\) – Ax – \\(B\\over x\\) = 0<\/p>\n

Thus, the function y = Ax\u00a0 + \\(B\\over x\\) satisfies the given equation.<\/p>\n\n\n

\n
Next – First Order First Degree Differential Equations<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Formation of Differential Equation<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find solution of differential equation i.e. general solution and particular solution with examples. Let’s begin – Solution of Differential Equation The solution of differential equation is a relation between the variables involved which satisfies the differential equation. for example, y = \\(e^x\\) is a solution of the differential equations …<\/p>\n

Solution of 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":[251,260,261,252],"yoast_head":"\nSolution of Differential Equation - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find solution of differential equation i.e. general solution and particular solution 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-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 Differential Equation - 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