{"id":217535,"date":"2025-05-22T05:10:42","date_gmt":"2025-05-22T05:10:42","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=217535"},"modified":"2025-05-22T05:10:44","modified_gmt":"2025-05-22T05:10:44","slug":"find-a-linear-differential-operator-that-annihilates-the-given-function-4","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/22\/find-a-linear-differential-operator-that-annihilates-the-given-function-4\/","title":{"rendered":"Find a linear differential operator that annihilates the given function"},"content":{"rendered":"\n

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

L(D)[f(x)]=0<\/p>\n\n\n\n

Where D=ddxD = \\frac{d}{dx}D=dxd\u200b is the differential operator.<\/p>\n\n\n\n

\n\n\n\n

Part 1: f(x)=1+6x\u22122x3f(x) = 1 + 6x – 2x^3f(x)=1+6x\u22122×3<\/strong><\/h3>\n\n\n\n

This is a polynomial of degree 3. Any polynomial of degree nnn can be annihilated by Dn+1D^{n+1}Dn+1.<\/p>\n\n\n\n

    \n
  • Here, the degree is 3.<\/li>\n\n\n\n
  • So, the annihilator is:<\/li>\n<\/ul>\n\n\n\n

    D4\\boxed{D^4}D4\u200b<\/p>\n\n\n\n

    Explanation:<\/p>\n\n\n\n

      \n
    • D[1+6x\u22122×3]=6\u22126x2D[1 + 6x – 2x^3] = 6 – 6x^2D[1+6x\u22122×3]=6\u22126×2<\/li>\n\n\n\n
    • D2[1+6x\u22122×3]=\u221212xD^2[1 + 6x – 2x^3] = -12xD2[1+6x\u22122×3]=\u221212x<\/li>\n\n\n\n
    • D3[1+6x\u22122×3]=\u221212D^3[1 + 6x – 2x^3] = -12D3[1+6x\u22122×3]=\u221212<\/li>\n\n\n\n
    • D4[1+6x\u22122×3]=0D^4[1 + 6x – 2x^3] = 0D4[1+6x\u22122×3]=0<\/li>\n<\/ul>\n\n\n\n

      Thus, the operator D4D^4D4 annihilates the polynomial function.<\/p>\n\n\n\n

      \n\n\n\n

      Part 2: f(x)=e\u2212x+2xex\u2212x2exf(x) = e^{-x} + 2x e^x – x^2 e^xf(x)=e\u2212x+2xex\u2212x2ex<\/strong><\/h3>\n\n\n\n

      This is a combination of two terms:<\/p>\n\n\n\n

        \n
      • e\u2212xe^{-x}e\u2212x<\/li>\n\n\n\n
      • (2x\u2212x2)ex(2x – x^2)e^x(2x\u2212x2)ex, which is a polynomial times exe^xex<\/li>\n<\/ul>\n\n\n\n

        We can treat each separately and take the least common multiple (LCM)<\/strong> of their annihilators.<\/p>\n\n\n\n

        For e\u2212xe^{-x}e\u2212x:<\/h4>\n\n\n\n
          \n
        • D+1D + 1D+1 annihilates e\u2212xe^{-x}e\u2212x because: (D+1)[e\u2212x]=D[e\u2212x]+e\u2212x=\u2212e\u2212x+e\u2212x=0(D + 1)[e^{-x}] = D[e^{-x}] + e^{-x} = -e^{-x} + e^{-x} = 0(D+1)[e\u2212x]=D[e\u2212x]+e\u2212x=\u2212e\u2212x+e\u2212x=0<\/li>\n<\/ul>\n\n\n\n

          For (2x\u2212x2)ex(2x – x^2)e^x(2x\u2212x2)ex:<\/h4>\n\n\n\n

          This is a polynomial of degree 2<\/strong> multiplied by exe^xex.<\/p>\n\n\n\n

            \n
          • A polynomial of degree 2 times exe^xex is annihilated by (D\u22121)3(D – 1)^3(D\u22121)3<\/li>\n<\/ul>\n\n\n\n

            Why?
            Because:<\/p>\n\n\n\n

              \n
            • If f(x)=P(x)exf(x) = P(x)e^xf(x)=P(x)ex, and P(x)P(x)P(x) is degree nnn, then the annihilator is (D\u22121)n+1(D – 1)^{n+1}(D\u22121)n+1<\/li>\n<\/ul>\n\n\n\n

              So:<\/p>\n\n\n\n

                \n
              • (D\u22121)3(D – 1)^3(D\u22121)3 annihilates (2x\u2212x2)ex(2x – x^2)e^x(2x\u2212x2)ex<\/li>\n<\/ul>\n\n\n\n

                Now take the LCM<\/strong> of the annihilators:<\/p>\n\n\n\n

                  \n
                • For e\u2212xe^{-x}e\u2212x: D+1D + 1D+1<\/li>\n\n\n\n
                • For (2x\u2212x2)ex(2x – x^2)e^x(2x\u2212x2)ex: (D\u22121)3(D – 1)^3(D\u22121)3<\/li>\n<\/ul>\n\n\n\n

                  So the overall annihilator is: (D+1)(D\u22121)3\\boxed{(D + 1)(D – 1)^3}(D+1)(D\u22121)3\u200b<\/p>\n\n\n\n

                  \n\n\n\n

                  \u2705 Final Answers:<\/h3>\n\n\n\n
                    \n
                  • For 1+6x\u22122×31 + 6x – 2x^31+6x\u22122×3<\/li>\n<\/ul>\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: L(D)[f(x)]=0 Where D=ddxD = \\frac{d}{dx}D=dxd\u200b is the differential operator. Part 1: f(x)=1+6x\u22122x3f(x) = 1 + 6x – 2x^3f(x)=1+6x\u22122×3 This is a polynomial of […]<\/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-217535","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217535","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=217535"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217535\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=217535"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=217535"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=217535"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}