{"id":32341,"date":"2022-09-22T23:48:38","date_gmt":"2022-09-22T15:48:38","guid":{"rendered":"https:\/\/ocw.nycu.edu.tw\/?post_type=course_page&#038;p=32341"},"modified":"2026-03-09T14:34:27","modified_gmt":"2026-03-09T06:34:27","slug":"deq031_141103","status":"publish","type":"course_page","link":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b-differential-equations-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e6%9d%8e%e6%a6%ae%e8%80%80%e8%80%81%e5%b8%ab\/deq031_141103","title":{"rendered":"Chapter 2: 1st order system and 2nd order equation"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"32341\" class=\"elementor elementor-32341\">\n\t\t\t\t\t\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-10 elementor-section elementor-top-section elementor-element elementor-element-16700d17 elementor-section-boxed elementor-section-height-default 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href=\"https:\/\/ocw.nycu.edu.tw\/\">Homepage<\/a><\/span><\/span><\/p>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-20 elementor-section elementor-top-section elementor-element elementor-element-6ad968e6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6ad968e6\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-52987d63\" data-id=\"52987d63\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e5808cb elementor-widget elementor-widget-heading\" data-id=\"e5808cb\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 *\/\n.elementor-heading-title{padding:0;margin:0;line-height:1}.elementor-widget-heading .elementor-heading-title[class*=elementor-size-]>a{color:inherit;font-size:inherit;line-height:inherit}.elementor-widget-heading .elementor-heading-title.elementor-size-small{font-size:15px}.elementor-widget-heading .elementor-heading-title.elementor-size-medium{font-size:19px}.elementor-widget-heading .elementor-heading-title.elementor-size-large{font-size:29px}.elementor-widget-heading .elementor-heading-title.elementor-size-xl{font-size:39px}.elementor-widget-heading .elementor-heading-title.elementor-size-xxl{font-size:59px}<\/style><h2 class=\"elementor-heading-title elementor-size-default\">Chapter 2: 1st order system and 2nd order equation\n<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7aec5b11 elementor-widget elementor-widget-video\" data-id=\"7aec5b11\" data-element_type=\"widget\" data-settings=\"{&quot;youtube_url&quot;:&quot;https:\\\/\\\/youtu.be\\\/Rn-x_4K_Iu0&quot;,&quot;modestbranding&quot;:&quot;yes&quot;,&quot;lazy_load&quot;:&quot;yes&quot;,&quot;video_type&quot;:&quot;youtube&quot;,&quot;controls&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 *\/\n.elementor-widget-video .elementor-widget-container{overflow:hidden;transform:translateZ(0)}.elementor-widget-video .elementor-wrapper{aspect-ratio:var(--video-aspect-ratio)}.elementor-widget-video .elementor-wrapper iframe,.elementor-widget-video .elementor-wrapper video{height:100%;width:100%;display:flex;border:none;background-color:#000}@supports not (aspect-ratio:1\/1){.elementor-widget-video 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class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-652aa46c elementor-widget elementor-widget-penci-button\" data-id=\"652aa46c\" data-element_type=\"widget\" data-widget_type=\"penci-button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"pcbtn-wrapper pcbtn-align-left\">\r\n\t\t\t<a class=\"pcbtn pcbtn-icon-left elementor-animation-grow\" href=\"\/?post_type=course_page&#038;p=29325\">\r\n\t\t\t\t<span class=\"pcbtn-wrapperin\">\r\n\t\t\t\t\t<span class=\"pcbtn-content\"><svg class=\"e-font-icon-svg e-fas-arrow-left\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M257.5 445.1l-22.2 22.2c-9.4 9.4-24.6 9.4-33.9 0L7 273c-9.4-9.4-9.4-24.6 0-33.9L201.4 44.7c9.4-9.4 24.6-9.4 33.9 0l22.2 22.2c9.5 9.5 9.3 25-.4 34.3L136.6 216H424c13.3 0 24 10.7 24 24v32c0 13.3-10.7 24-24 24H136.6l120.5 114.8c9.8 9.3 10 24.8.4 34.3z\"><\/path><\/svg>\u8fd4\u56de\u8ab2\u7a0b\u9996\u9801<\/span>\r\n\t\t\t\t\t\t\t\t\t<\/span>\r\n\t\t\t<\/a>\r\n\t\t<\/div>\r\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-2 penci-sticky-ct    elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-fc8fc3b\" data-id=\"fc8fc3b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<div class=\"elementor-element elementor-element-546b36ef elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"546b36ef\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 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100%;-webkit-mask-repeat:var(--divider-pattern-repeat);mask-repeat:var(--divider-pattern-repeat);background-color:var(--divider-color);-webkit-mask-image:var(--divider-pattern-url);mask-image:var(--divider-pattern-url)}.elementor-widget-divider--no-spacing{--divider-pattern-size:auto}.elementor-widget-divider--bg-round{--divider-pattern-repeat:round}.rtl .elementor-widget-divider .elementor-divider__text{direction:rtl}.e-con-inner>.elementor-widget-divider,.e-con>.elementor-widget-divider{width:var(--container-widget-width,100%);--flex-grow:var(--container-widget-flex-grow)}<\/style>\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-da19150 elementor-widget elementor-widget-shortcode\" data-id=\"da19150\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\">\n<table id=\"tablepress-64\" class=\"tablepress tablepress-id-64\">\n<thead>\n<tr class=\"row-1 odd\">\n\t<th class=\"column-1\">\u9031\u6b21<\/th><th class=\"column-2\">\u8ab2\u7a0b\u5167\u5bb9<\/th><th class=\"column-3\">\u8ab2\u7a0b\u5f71\u97f3<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-hover\">\n<tr class=\"row-2 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u8ab2\u7a0b\u4ecb\u7d39<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=29361\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-3 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Introduction of ODE<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32243\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-4 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Big Picture for ODE<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32257\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-5 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">1st order linear system<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32263\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-6 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">1st order linear system: \u91cf\u7684\u5206\u6790<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32269\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-7 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">2nd order linear equation with constant coefficients<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32277\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-8 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Practical Problem<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32283\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-9 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Two more \u201cderivation\u201d of math models from:<br \/>\n--Predator-Prey problem<br \/>\n--Deposit and Withdraw problem<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32289\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-10 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">1.2 ODE Big Picture<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32297\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-11 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">1.2 Mixed Problem<br \/>\n1.3 Slope Field<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32303\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-12 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">1.3 Slope Field<br \/>\n1.4 Numerical Technique<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32313\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-13 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">1.5 IVP\u89e3: \u5b58\u5728\u6027\u8207\u552f\u4e00\u6027<br \/>\n1.6 \u8cea\u5206\u6790, phase line(for autonomous system)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32323\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-14 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">1.7 Bifurcation and one supplement for 1.6(real problem)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32329\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-15 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Complete Chapter 1<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32335\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-16 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Chapter 2: 1st order system and 2nd order equation<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32341\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-17 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">2.2 Geometry of the system(by vector field)<br \/>\n2.3 Damped harmonic oscillator<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32347\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-18 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">2.4 Special systems(solutions can be solved explicitly)<br \/>\n2.5 Euler\u2019s method for 1st order system<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32353\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-19 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">2.6 \u5b58\u5728\u8207\u552f\u4e00\u6027(for 1st order system)<br \/>\n2.7 (\u5047\u7684) 3D-system (\u6d41\u884c\u75c5\u50b3\u67d3\u554f\u984c)<br \/>\n2.8 (\u771f\u7684) 3D-system (\u6c23\u8c61\u9810\u5831\u7c21\u5316\u7cfb\u7d71: Lorenz equation)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32359\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-20 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">3.1 \u7dda\u6027\u7406\u8ad6<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32365\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-21 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">3.2 Straight-Line Solutions<br \/>\n3.3 Phase Portraits for Linear Systems with Real Eigenvalues<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32371\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-22 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">3.2 Straight-Line Solutions<br \/>\n3.3 Phase Portraits for Linear Systems with Real Eigenvalues<br \/>\n3.4 complex valued eigenvalues<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32377\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-23 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">3.5 Special cases of Linear systems<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32383\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-24 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">To complete 3.5<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32393\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-25 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">3.5 to complete<br \/>\n3.6 2nd order linear system and 2D linear system<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32399\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-26 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">To complete 3.6<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32405\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-27 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">3.7 Trace \u2013 Determinant Plane<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32411\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-28 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">To complete chapter 3<br \/>\n3.7 Part II: T-D plane<br \/>\n3.8 3D linear system<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32417\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-29 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Non-linear 2nd order ode:<br \/>\n--Pendulum motion<br \/>\n--Pendulum motion and spring motion<br \/>\n--Differences of phase portraits between linear and Nonlinear 2nd order equations<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32423\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-30 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Continue nonlinear 2nd order ode<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32429\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-31 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Special case for solving problems<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=32435\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-64 from cache --><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-3 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-5f099062\" data-id=\"5f099062\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap 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