{"id":2901,"date":"2024-01-03T22:15:23","date_gmt":"2024-01-03T21:15:23","guid":{"rendered":"https:\/\/mxth.dk\/?page_id=2901"},"modified":"2024-03-06T21:21:01","modified_gmt":"2024-03-06T20:21:01","slug":"opgaver-til-differentialligninger","status":"publish","type":"page","link":"https:\/\/mxth.dk\/?page_id=2901","title":{"rendered":"Opgaver til differentialligninger"},"content":{"rendered":"\n<p class=\" eplus-wrapper\"><\/p>\n\n\n\n<div class=\"wp-block-group eplus-wrapper is-layout-constrained wp-block-group-is-layout-constrained\"><p class=\" eplus-wrapper eplus-styles-uid-dde4ef\"><strong>Introduktion til differentialligninger<\/strong><\/p>\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(y\u2019=f(x)\\)<\/p>\n\n\n\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-id2901_394c11-55 kt-accordion-has-15-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"false\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-10 kt-pane2901_7f468c-69\" id=\"IDL001\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">IDL001<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Bestem den fuldst\u00e6ndige l\u00f8sning til f\u00f8lgende differentialligning \\(\\frac{d}{dx}f=4\\cdot\\sin(x)\\).<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-11 kt-pane2901_b42277-89\" id=\"IDL002\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">IDL002<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Bestem en l\u00f8sning til f\u00f8lgende ligninger og betingelser:<\/p>\n\n\n<ol class=\"eplus-wrapper wp-block-list eplus-styles-uid-da54e3\">\n<li class=\" eplus-wrapper\">\\(y\u2019=0,\\hspace{1cm} y(-2)=3\\)<\/li>\n\n\n\n<li class=\" eplus-wrapper\">\\(y\u2019=x,\\hspace{1cm} y(3)=7\\)<\/li>\n\n\n\n<li class=\" eplus-wrapper\">\\(y\u2019=4x-3,\\hspace{1cm} y(-3)=3\\)<\/li>\n<\/ol><\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-12 kt-pane2901_fbef33-88\" id=\"IDL003\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">IDL003<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Du skal l\u00f8se f\u00f8lgende differentialligninger<\/p>\n\n\n<ol class=\"eplus-wrapper wp-block-list eplus-styles-uid-a8b052\">\n<li class=\" eplus-wrapper\">\\(y\u2019=-0,\\!5\\cdot x^3+0,\\!25\\cdot x^2-4\\cdot x +5\\)\u00a0<\/li>\n\n\n\n<li class=\" eplus-wrapper\">\\(3\\cdot x^2 + f\u2019(x)=-4\\cdot x\\)<\/li>\n\n\n\n<li class=\" eplus-wrapper\">\\(17-9\\cdot x^2+\\frac{dy}{dx}=14\\cdot x\\)<\/li>\n<\/ol><\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-13 kt-pane2901_2ef168-ca\" id=\"IDL004\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">IDL004<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Bestem den fuldst\u00e6ndige l\u00f8sning til f\u00f8lgende differentialligning \\(-3\\cdot f\u2019(x)=x^{-4}+7\\).<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-14 kt-pane2901_94ca9a-4c\" id=\"IDL005\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">IDL005<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Bestem den fuldst\u00e6ndige l\u00f8sning til f\u00f8lgende differentialligning \\(\\frac{\\frac{d}{dt}f(t)}{t}=3\\cdot t+7\\).<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-15 kt-pane2901_6e7a49-18\" id=\"IDL006\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">IDL006<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Du skal l\u00f8se f\u00f8lgende differentialligning \\(y\u2019=0,\\!5\\cdot x^2\\cdot\\sqrt{4x^3-8}\\).<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-group eplus-wrapper is-layout-constrained wp-block-group-is-layout-constrained\"><p class=\" eplus-wrapper eplus-styles-uid-dde4ef\"><strong>V\u00e6kstens afh\u00e6ngighed af populationen st\u00f8rrelsen<\/strong><\/p>\n\n\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-id2901_86c51d-54 kt-accordion-has-15-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"false\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-10 kt-pane2901_9dc5d4-76\" id=\"VAAPS001\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VAAPS001<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Antag, at energiniveauet \\(E(t)\\) i en elektronisk enhed falder proportionalt med forskellen mellem dens aktuelle energiniveau og det minimale acceptable niveau \\(E_{min}\\). Differentialligningen kan udtrykkes som \\(E\u2019(t)=-k\\cdot (E(t)-E_{min})\\), hvor \\(k\\) er en positiv konstant.<\/p>\n\n\n<ol class=\"eplus-wrapper wp-block-list eplus-styles-uid-716976\">\n<li class=\" eplus-wrapper\">Find den generelle l\u00f8sning af differentialligningen.<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Antag, at den elektroniske enhed starter med et energiniveau p\u00e5 75 J og har et minimalt acceptabelt niveau p\u00e5 33 J. Efter 100 sekunder er energiniveauet 72,55 J. Bestem konstanten \\(k\\).<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Tolk l\u00f8sningen i forhold til, hvordan eneginiveauet udvikler sig over tid og hvordan konstanten \\(k\\) p\u00e5virker denne udvikling.<\/li>\n<\/ol><\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-11 kt-pane2901_c10840-d8\" id=\"VAAPS002\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VAAPS002<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Antag, at antallet af digitale enheder i en befolkning, \\(P(t)\\), vokser med en konstant sats \\(r\\).<\/p>\n\n\n<ol class=\"eplus-wrapper wp-block-list eplus-styles-uid-29a964\">\n<li class=\" eplus-wrapper\">Formuler differentialligningen, der beskriver v\u00e6ksten af digitale enheder.<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Find den fuldst\u00e6ndige l\u00f8sning til differentialligningen.<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Antag, at antallet af digital enheder starter med 340.000 enheder. Bestem v\u00e6kstraten \\(r\\) n\u00e5r det vides at til tiden 7 er der 476.000 enheder.<\/li>\n<\/ol><\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-12 kt-pane2901_4f496d-78\" id=\"VAAPS003\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VAAPS003<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Antag, at temperaturen \\(T(t)\\) af en computerprocessor \u00e6ndre proportionalt med forskellen p\u00e5 dens aktuelle temperatur og den maksimale sikkerhedstemperatur \\(T_{max}\\). <\/p>\n\n\n<ol class=\"eplus-wrapper wp-block-list eplus-styles-uid-f56753\">\n<li class=\" eplus-wrapper\">Formuler differentialligningen, der beskriver temperaturkontrollen i computerprocessoren.<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Find den fuldst\u00e6ndige l\u00f8sning til differentialligningen.<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Antag, at processoren starter ved en temperatur p\u00e5 \\(27^{circ}C\\). Bestem konstanten k n\u00e5r det vides at sikkerhedstemperaturen er \\(73^{\\circ}C\\) og at til tiden 4,3 er temperaturen \\(31^{\\circ}C\\).<\/li>\n<\/ol><\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-13 kt-pane2901_0d388e-28\" id=\"VAAPS004\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VAAPS004<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">En pr\u00f8ve af et radioaktivt stof indeholder \\(N_0\\) enheder ved tiden \\(t=0\\). \u00c6ndringen af radioaktivt stof er proportionalt med m\u00e6ngden til et givent tidspunkt. <\/p>\n\n\n<ol class=\"eplus-wrapper wp-block-list eplus-styles-uid-047564\">\n<li class=\" eplus-wrapper\">Opstil en differentialligning for m\u00e6ngden af radioaktivt stof i pr\u00f8ven.<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Find den fuldst\u00e6ndige l\u00f8sning til differentialligningen.<\/li>\n\n\n\n<li class=\" eplus-wrapper\">For Barium-137 er halveringstiden 2,55 min. Find den partikul\u00e6rer l\u00f8sning n\u00e5r det oplyses at l\u00f8sningenskurven g\u00e5r igennem tiden 43 min er antallet af barium-137 654,5 og \u00e6ndringen er 177,9. <\/li>\n\n\n\n<li class=\" eplus-wrapper\">Fortolk l\u00f8sningen, n\u00e5r det oplyses at der i en ny ris\u00f8-kilde er ca. 78 mio barium-137. Hvordan p\u00e5virker konstanten \\(k\\) radioaktiviteten?<\/li>\n<\/ol><\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-14 kt-pane2901_8e3ecf-4a\" id=\"VAAPS005\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VAAPS005<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">En bakteriekultur starter med 2300 bakterier og vokser med en konstant v\u00e6kstrate. V\u00e6sktraten kan beskrives med formlen <\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(k={\\ln (2)\\over T_{1\/2}}\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Bakteriekulturen fordobles hver 20 min.<\/p>\n\n\n<ol class=\"eplus-wrapper wp-block-list eplus-styles-uid-8a42da\">\n<li class=\" eplus-wrapper\">Bestem den fuldst\u00e6ndige l\u00f8sning til differentialligningen der beskriver  bakteriekulturen.<\/li>\n<\/ol><\/div><\/div><\/div>\n<\/div><\/div><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Introduktion til differentialligninger \\(y\u2019=f(x)\\) V\u00e6kstens afh\u00e6ngighed af populationen st\u00f8rrelsen<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"ub_ctt_via":"","editor_plus_copied_stylings":"{}","footnotes":""},"categories":[],"tags":[],"class_list":["post-2901","page","type-page","status-publish","hentry"],"featured_image_src":null,"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/pages\/2901","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2901"}],"version-history":[{"count":10,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/pages\/2901\/revisions"}],"predecessor-version":[{"id":3105,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/pages\/2901\/revisions\/3105"}],"wp:attachment":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2901"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2901"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2901"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}