{"id":815,"date":"2025-10-09T15:40:50","date_gmt":"2025-10-09T15:40:50","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=815"},"modified":"2025-12-16T16:47:05","modified_gmt":"2025-12-16T16:47:05","slug":"linear-homogeneous-equations-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/linear-homogeneous-equations-apply-it\/","title":{"raw":"Linear Homogeneous Equations: Apply It","rendered":"Linear Homogeneous Equations: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify homogeneous linear equations<\/li>\r\n \t<li>Find and use characteristic equations<\/li>\r\n \t<li>Solve homogeneous equations using characteristic equations<\/li>\r\n \t<li>Solve initial and boundary value problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Analyzing Sound Wave Equations in a Synthesizer App<\/h2>\r\nYou are helping a friend develop a simple synthesizer app that generates different types of sound waves. The app models audio signals using second-order differential equations, where the solution [latex]y(t) [\/latex] represents the amplitude of the sound wave at time [latex]t[\/latex].\u00a0Different equation settings produce different audio behaviors: some create sustained tones, others create sounds that fade out, and some produce unusual effects. Your task is to analyze several differential equations to determine what type of sound each setting will produce. You will classify equations, find characteristic equations, solve for general solutions, and use initial conditions to find specific waveforms. Understanding these mathematical relationships will help your friend's app generate the desired audio output.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317439[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317440[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317441[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317442[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317443[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify homogeneous linear equations<\/li>\n<li>Find and use characteristic equations<\/li>\n<li>Solve homogeneous equations using characteristic equations<\/li>\n<li>Solve initial and boundary value problems<\/li>\n<\/ul>\n<\/section>\n<h2>Analyzing Sound Wave Equations in a Synthesizer App<\/h2>\n<p>You are helping a friend develop a simple synthesizer app that generates different types of sound waves. The app models audio signals using second-order differential equations, where the solution [latex]y(t)[\/latex] represents the amplitude of the sound wave at time [latex]t[\/latex].\u00a0Different equation settings produce different audio behaviors: some create sustained tones, others create sounds that fade out, and some produce unusual effects. Your task is to analyze several differential equations to determine what type of sound each setting will produce. You will classify equations, find characteristic equations, solve for general solutions, and use initial conditions to find specific waveforms. Understanding these mathematical relationships will help your friend&#8217;s app generate the desired audio output.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317439\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317439&theme=lumen&iframe_resize_id=ohm317439&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317440\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317440&theme=lumen&iframe_resize_id=ohm317440&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317441\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317441&theme=lumen&iframe_resize_id=ohm317441&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317442\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317442&theme=lumen&iframe_resize_id=ohm317442&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317443\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317443&theme=lumen&iframe_resize_id=ohm317443&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":10,"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-815","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\/815","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\/815\/revisions"}],"predecessor-version":[{"id":2811,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/815\/revisions\/2811"}],"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\/815\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=815"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=815"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=815"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=815"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}