Find a linear differential operator that annihilates the given function. (Use D for the differential operator.)<\/p>\n\n\n\n
For, 1+6x – 2×3 and, e-x + 2xex – x2ex<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n To find a linear differential operator ( L(D) ) that annihilates a given function, we look for a differential equation of the form:<\/p>\n\n\n\n [ where ( L(D) ) is a polynomial in the differentiation operator ( D ) (where ( D = \\frac{d}{dx} )), and ( y(x) ) is the given function.<\/p>\n\n\n\n This is a polynomial function. The general rule is that a polynomial of degree ( n ) is annihilated by ( D^{(n+1)} ). Here, the highest degree term is ( -2x^3 ), so we use:<\/p>\n\n\n\n [ Thus, the annihilating operator is:<\/p>\n\n\n\n [ This function consists of terms involving exponentials and polynomials. The general rule is:<\/p>\n\n\n\n In our case:<\/p>\n\n\n\n Thus, the minimal annihilating operator for ( g(x) ) is the least common multiple (LCM) of these:<\/p>\n\n\n\n [ A differential operator<\/strong> annihilates a function when applying the operator results in zero. For polynomials, the order of the derivative required is one more than the polynomial degree. For exponentials, a first-order differential operator suffices, but if multiplied by ( x^n ), we need ( (D – \\lambda)^{n+1} ). By finding the LCM of all required factors, we determine the minimal operator.<\/p>\n","protected":false},"excerpt":{"rendered":" Find a linear differential operator that annihilates the given function. (Use D for the differential operator.) For, 1+6x – 2×3 and, e-x + 2xex – x2ex The correct answer and explanation is : To find a linear differential operator ( L(D) ) that annihilates a given function, we look for a differential equation of the […]<\/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-204079","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/204079","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=204079"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/204079\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=204079"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=204079"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=204079"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
L(D) y = 0
]<\/p>\n\n\n\n1st Function: ( f(x) = 1 + 6x – 2x^3 )<\/h3>\n\n\n\n
D^{4} (1 + 6x – 2x^3) = 0
]<\/p>\n\n\n\n
L(D) = D^4
]<\/p>\n\n\n\n
\n\n\n\n2nd Function: ( g(x) = e^{-x} + 2x e^x – x^2 e^x )<\/h3>\n\n\n\n
\n
\n
L(D) = (D + 1)(D – 1)^3
]<\/p>\n\n\n\n
\n\n\n\nExplanation:<\/h3>\n\n\n\n