{"id":154480,"date":"2022-03-25T23:34:56","date_gmt":"2022-03-25T18:04:56","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/variation-of-parameters-explanation-methods-and-solved-examples\/"},"modified":"2023-01-02T14:34:11","modified_gmt":"2023-01-02T09:04:11","slug":"variation-of-parameters-explanation-methods-and-solved-examples","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/variation-of-parameters\/","title":{"rendered":"Variation of Parameters &#8211; Explanation, Methods and Solved Examples"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/variation-of-parameters\/#What_is_Variation_in_Maths\" title=\"What is Variation in Maths?\">What is Variation in Maths?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/variation-of-parameters\/#Variation_of_Parameters\" title=\"Variation of Parameters\">Variation of Parameters<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/variation-of-parameters\/#Method_of_Variation_of_Parameters\" title=\"Method of Variation of Parameters\">Method of Variation of Parameters<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/variation-of-parameters\/#Two_Methods_in_Variation_of_Parameters\" title=\"Two Methods in Variation of Parameters\">Two Methods in Variation of Parameters<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/variation-of-parameters\/#Solutions_to_Variation_of_Parameters\" title=\"Solutions to Variation of Parameters\">Solutions to Variation of Parameters<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/variation-of-parameters\/#Solved_Example_using_Variation_of_Parameter_Formula\" title=\"Solved Example using Variation of Parameter Formula\">Solved Example using Variation of Parameter Formula<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_is_Variation_in_Maths\"><\/span>What is Variation in Maths?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Variation is a measure of how much one quantity changes in relation to another quantity. It is used to find out how much two quantities vary from each other.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-154479 size-medium\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/variation-of-parameters-explanation-methods-and-solved-examples-300x212.jpg\" alt=\"Variation of Parameters \" width=\"300\" height=\"212\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/variation-of-parameters-explanation-methods-and-solved-examples-300x212.jpg?v=1648231490 300w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/variation-of-parameters-explanation-methods-and-solved-examples.jpg?v=1648231490 606w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Variation_of_Parameters\"><\/span>Variation of Parameters<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The variation of parameters method is a numerical technique used to solve differential equations. It is a variation of the Runge-Kutta method.<\/p>\n<p>The technique works by varying the parameters of the differential equation in order to find a solution that best matches the given data.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Method_of_Variation_of_Parameters\"><\/span>Method of Variation of Parameters<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The method of variation of parameters is a technique used to find the solution to a differential equation. The method uses the initial conditions of the equation to find a series of solutions. Each solution is a function of the parameter being varied.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Two_Methods_in_Variation_of_Parameters\"><\/span>Two Methods in Variation of Parameters<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The first method is the direct substitution method. In this method, the parameter is varied and the resulting equation is solved.<\/p>\n<p>The second method is the variation of parameters method. In this method, the parameter is varied and the resulting equations are solved simultaneously.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Solutions_to_Variation_of_Parameters\"><\/span>Solutions to Variation of Parameters<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The solutions to variation of parameters are found by taking the derivative of the original equation with respect to each parameter and setting it equal to zero. This will result in a set of equations that can be solved for the unknowns in the equation.<\/p>\n<p>1. First, take the derivative of the original equation with respect to time.<\/p>\n<p>d[itex]\\left(\\frac{dx}{dt}\\right)[\/itex]\n<p>= [itex]\\frac{d}{dt}x[\/itex]\n<p>= [itex]\\frac{d}{dt}(x^{2}+y^{2})[\/itex]\n<p>Next, set this derivative equal to zero and solve for time.<\/p>\n[itex]\\frac{d}{dt}x^{2}+y^{2}[\/itex]\n<p>= 0<\/p>\n[itex]dt[\/itex]\n<p>= [itex]\\frac{-2x}{y^{2}}[\/itex]\n<p>2. First, take the derivative of the original equation with respect to x.<\/p>\n<p>d[itex]\\left(\\frac{dx}{dt}\\right)[\/itex]\n<p>= [itex]\\frac{d}{dt}x[\/itex]\n<p>= [itex]\\frac{d}{dt}(x^{2}+y^{2})[\/ite<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Solved_Example_using_Variation_of_Parameter_Formula\"><\/span>Solved Example using Variation of Parameter Formula<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The displacement of a harmonic oscillator is given by<\/p>\n<p>x(t) = A sin(\u03c9t + \u03c6)<\/p>\n<p>where A is the amplitude, \u03c9 is the angular frequency, and \u03c6 is the phase.<\/p>\n<p>What is the displacement at time t = 0?<\/p>\n<p>x(0) = A sin(\u03c90t + \u03c6)<\/p>\n<p>x(0) = A sin(\u03c90t + \u03c6)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What is Variation in Maths? Variation is a measure of how much one quantity changes in relation to another quantity. 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