Solve the Differential Equation (12x+5y-9)dx+(5x-2y-4)dy=0<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n The given differential equation is:<\/p>\n\n\n\n [ To solve this first-order differential equation, we will determine if it is exact. A differential equation of the form ( M(x, y) \\, dx + N(x, y) \\, dy = 0 ) is exact if the partial derivatives of ( M(x, y) ) with respect to ( y ) and ( N(x, y) ) with respect to ( x ) are equal:<\/p>\n\n\n\n [ From the given equation:<\/p>\n\n\n\n We now calculate the partial derivatives:<\/p>\n\n\n\n Since the partial derivatives are equal, the equation is exact.<\/p>\n\n\n\n For an exact equation, there exists a potential function ( \\Psi(x, y) ) such that:<\/p>\n\n\n\n [ Integrate ( M(x, y) = 12x + 5y – 9 ) with respect to ( x ):<\/p>\n\n\n\n [ where ( h(y) ) is an arbitrary function of ( y ) (since the derivative with respect to ( x ) does not affect ( y )).<\/p>\n\n\n\n Now, take the partial derivative of ( \\Psi(x, y) ) with respect to ( y ):<\/p>\n\n\n\n [ We know that ( \\frac{\\partial \\Psi}{\\partial y} = N(x, y) = 5x – 2y – 4 ). Therefore, we have the equation:<\/p>\n\n\n\n [ Canceling out ( 5x ) from both sides:<\/p>\n\n\n\n [ Integrating with respect to ( y ):<\/p>\n\n\n\n [ where ( C ) is a constant of integration.<\/p>\n\n\n\n Thus, the potential function is:<\/p>\n\n\n\n [ The general solution to the differential equation is obtained by setting ( \\Psi(x, y) = C_1 ), where ( C_1 ) is a constant:<\/p>\n\n\n\n [ This is the implicit solution to the differential equation.<\/p>\n","protected":false},"excerpt":{"rendered":" Solve the Differential Equation (12x+5y-9)dx+(5x-2y-4)dy=0 The correct answer and explanation is : The given differential equation is: [(12x + 5y – 9) \\, dx + (5x – 2y – 4) \\, dy = 0] To solve this first-order differential equation, we will determine if it is exact. A differential equation of the form ( M(x, […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","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":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-209161","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/209161","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=209161"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/209161\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=209161"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=209161"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=209161"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
(12x + 5y – 9) \\, dx + (5x – 2y – 4) \\, dy = 0
]<\/p>\n\n\n\n
\\frac{\\partial M}{\\partial y} = \\frac{\\partial N}{\\partial x}
]<\/p>\n\n\n\nStep 1: Define the functions ( M(x, y) ) and ( N(x, y) )<\/h3>\n\n\n\n
\n
Step 2: Check if the equation is exact<\/h3>\n\n\n\n
\n
Step 3: Solve the equation<\/h3>\n\n\n\n
\\frac{\\partial \\Psi}{\\partial x} = M(x, y) \\quad \\text{and} \\quad \\frac{\\partial \\Psi}{\\partial y} = N(x, y)
]<\/p>\n\n\n\nStep 3a: Find ( \\Psi(x, y) ) from ( \\frac{\\partial \\Psi}{\\partial x} = M(x, y) )<\/h4>\n\n\n\n
\\Psi(x, y) = \\int (12x + 5y – 9) \\, dx = 6x^2 + 5xy – 9x + h(y)
]<\/p>\n\n\n\nStep 3b: Find ( h(y) ) using ( \\frac{\\partial \\Psi}{\\partial y} = N(x, y) )<\/h4>\n\n\n\n
\\frac{\\partial \\Psi}{\\partial y} = 5x + h'(y)
]<\/p>\n\n\n\n
5x + h'(y) = 5x – 2y – 4
]<\/p>\n\n\n\n
h'(y) = -2y – 4
]<\/p>\n\n\n\n
h(y) = -y^2 – 4y + C
]<\/p>\n\n\n\nStep 3c: Combine the results<\/h4>\n\n\n\n
\\Psi(x, y) = 6x^2 + 5xy – 9x – y^2 – 4y + C
]<\/p>\n\n\n\nStep 4: Set the potential function equal to a constant<\/h3>\n\n\n\n
6x^2 + 5xy – 9x – y^2 – 4y = C_1
]<\/p>\n\n\n\n