{"id":825,"date":"2025-10-09T15:41:51","date_gmt":"2025-10-09T15:41:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=825"},"modified":"2025-12-16T17:35:23","modified_gmt":"2025-12-16T17:35:23","slug":"nonhomogeneous-linear-equations-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/nonhomogeneous-linear-equations-apply-it\/","title":{"raw":"Nonhomogeneous Linear Equations: Apply It","rendered":"Nonhomogeneous Linear Equations: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find general solutions of nonhomogeneous equations<\/li>\r\n \t<li>Apply the method of undetermined coefficients<\/li>\r\n \t<li>Apply the method of variation of parameters<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Modeling Food Truck Profit with Nonhomogeneous Equations<\/h2>\r\nYour friend recently started running a small food truck near campus, and they've asked for your help analyzing their daily profit trends. After collecting data for several weeks, you notice that the profit rate changes over time due to two main factors: a natural settling toward a baseline as regular customers establish habits, and external boosts from campus events that drive additional sales. The profit behavior can be modeled using a second-order linear differential equation where the right-hand side represents the external influences on revenue. Your task is to help your friend understand both the long-term natural behavior of their profits and how specific events create temporary changes. By finding the general solution to this nonhomogeneous equation, you can help predict future earnings and plan for busy periods.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317444[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317445[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317446[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317447[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317448[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find general solutions of nonhomogeneous equations<\/li>\n<li>Apply the method of undetermined coefficients<\/li>\n<li>Apply the method of variation of parameters<\/li>\n<\/ul>\n<\/section>\n<h2>Modeling Food Truck Profit with Nonhomogeneous Equations<\/h2>\n<p>Your friend recently started running a small food truck near campus, and they&#8217;ve asked for your help analyzing their daily profit trends. After collecting data for several weeks, you notice that the profit rate changes over time due to two main factors: a natural settling toward a baseline as regular customers establish habits, and external boosts from campus events that drive additional sales. The profit behavior can be modeled using a second-order linear differential equation where the right-hand side represents the external influences on revenue. Your task is to help your friend understand both the long-term natural behavior of their profits and how specific events create temporary changes. By finding the general solution to this nonhomogeneous equation, you can help predict future earnings and plan for busy periods.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317444\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317444&theme=lumen&iframe_resize_id=ohm317444&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317445\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317445&theme=lumen&iframe_resize_id=ohm317445&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317446\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317446&theme=lumen&iframe_resize_id=ohm317446&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317447\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317447&theme=lumen&iframe_resize_id=ohm317447&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317448\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317448&theme=lumen&iframe_resize_id=ohm317448&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":15,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-825","chapter","type-chapter","status-publish","hentry"],"part":440,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/825","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/825\/revisions"}],"predecessor-version":[{"id":2813,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/825\/revisions\/2813"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/440"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/825\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=825"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=825"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=825"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=825"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}