{"page":"\u003clink rel=\"stylesheet\" href=\"https://lessonplanet.com/assets/packs/css/resources-c03aa079.css\" /\u003e\n\u003clink rel=\"stylesheet\" href=\"https://lessonplanet.com/assets/packs/css/lp_boclips_stylesheets-517835be.css\" media=\"all\" /\u003e\n\u003cdiv data-title='A better way to understand Differential Equations | Nonlinear Dynamics (Part 3)' data-url='/boclips/videos/628c91d5408eb327c3205701' data-video-url='/boclips/videos/628c91d5408eb327c3205701' id='bo_player_modal'\u003e\n\u003cdiv class='boclips-resource-page modal-dialog panel-container'\u003e\n\u003cdiv class='react-notifications-root'\u003e\u003c/div\u003e\n\u003cdiv class='rp-header'\u003e\n\u003cdiv class='rp-type'\u003e\n\u003ci aria-hidden='true' class='fai fa-regular fa-circle-play'\u003e\u003c/i\u003e\nVideo\n\u003c/div\u003e\n\u003ch1 class='rp-title' id='video-title'\u003e\nA better way to understand Differential Equations | Nonlinear Dynamics (Part 3)\n\u003c/h1\u003e\n\u003cdiv class='rp-actions'\u003e\n\u003cdiv class='mr-1'\u003e\n\u003ca class=\"btn btn-success\" data-posthog-event=\"Signup: LP Signup Activity\" data-posthog-location=\"body_link_boclips\" data-remote=\"true\" href=\"/subscription/new\"\u003e\u003cspan\u003e\u003cspan\u003eGet Free Access\u003c/span\u003e\u003cspan class=\"\"\u003e for 10 Days\u003c/span\u003e\u003cspan\u003e!\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv class='rp-body'\u003e\n\u003cdiv class='rp-info'\u003e\n\u003cdiv aria-label='Hide resource details' class='rp-hide-info' role='button' tabindex='0'\u003e\u0026times;\u003c/div\u003e\n\u003ci aria-label='Expand resource details' class='rp-expand-info fai fa-solid fa-up-right-and-down-left-from-center' role='button' tabindex='0'\u003e\u003c/i\u003e\n\u003ci aria-label='Compress resource details' class='rp-compress-info fai fa-solid fa-down-left-and-up-right-to-center' role='button' tabindex='0'\u003e\u003c/i\u003e\n\u003cdiv class='rp-rating'\u003e\n\u003cspan class='resource-pool'\u003e\n\u003cspan class='pool-label'\u003ePublisher:\u003c/span\u003e\n\u003cspan class='pool-name'\u003e\n\u003cspan class='text'\u003e\u003ca data-publisher-id=\"30360345\" href=\"/search?publisher_ids%5B%5D=30360345\"\u003eVirtually Passed\u003c/a\u003e\u003c/span\u003e\n\u003c/span\u003e\n\u003c/span\u003e\n\u003c/div\u003e\n\u003cdiv class='rp-description'\u003e\n\u003cspan class='short-description'\u003eThese second-order nonlinear differential equations can be written in the form:dx/dt = f(x,y)dy/dt = g(x,y)Got a nonlinear differential equation? No problem, just linearize it! This method approximates the vector field as a linear...\u003c/span\u003e\n\u003cspan class='full-description hide'\u003eThese second-order nonlinear differential equations can be written in the form:\u003cbr/\u003edx/dt = f(x,y)\u003cbr/\u003edy/dt = g(x,y)\u003cbr/\u003e\u003cbr/\u003eGot a nonlinear differential equation? No problem, just linearize it! This method approximates the vector field as a linear equation at fixed points. This is very helpful for understanding stability and giving some qualitative ideas about the 'big picture' dynamics. There are draw-backs to linearization though. Firstly, it only illuminates the flow around the fixed points. Secondly, special types of fixed points that weren't covered in this series (centers, stars etc) cannot be reliably predicted using this method.\u003cbr/\u003e\u003cbr/\u003eChapters:\u003cbr/\u003e0:00 Intro \u0026 Recap\u003cbr/\u003e0:31 Motivation\u003cbr/\u003e1:14 Damped Pendulum Equation of Motion\u003cbr/\u003e2:20 Damped Pendulum Phase Plane Generation\u003cbr/\u003e3:02 Damped Pendulum is Nonlinear\u003cbr/\u003e3:32 1D flow Linearization Motivation\u003cbr/\u003e3:57 2D flow Linearization Maths\u003cbr/\u003e4:29 Damped Pendulum Phase Plane Fixed Points Classification\u003cbr/\u003e5:44 Stable Spiral Initial Condition\u003cbr/\u003e5:55 Saddle Initial Condition\u003cbr/\u003e6:09 Van der Pol Phase Plane\u003cbr/\u003e6:38 Van der Pol Linearization\u003cbr/\u003e7:09 Rambling about Nonlinear Differential Equations\u003cbr/\u003e7:47 Outro\u003c/span\u003e\n\u003c/div\u003e\n\u003cdiv class='action-container flex justify-between'\u003e\n\u003cbutton aria-expanded='false' 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