{"id":108,"date":"2019-06-10T16:11:58","date_gmt":"2019-06-10T20:11:58","guid":{"rendered":"https:\/\/blogs.harvard.edu\/siams\/?p=108"},"modified":"2019-06-10T17:21:29","modified_gmt":"2019-06-10T21:21:29","slug":"dynamical-systems-math-1b-differential-equations-background","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-1b-differential-equations-background\/","title":{"rendered":"Dynamical systems: Math 1b differential equations background."},"content":{"rendered":"<p>I have been using the Strogatz textbook for teaching dynamical systems. \u00a0The course has multivariable calculus and linear algebra prerequisites. \u00a0Students might take the prerequisite courses different places. \u00a0For students who have taken Math 1b, AM\/Math 21a, Math 21b, there was 6-7 week of differential equations background (11 classes in Math 1b + 9 classes in 21b).<\/p>\n<p>Student diff eq background from Math 1b:<\/p>\n<ul>\n<li>Intro to diff eq\n<ul>\n<li>Write equations from a description of a model where the rate of change of a variable depends on the variable.<\/li>\n<li>Determine whether a provided function is a solution to a differential equation.<\/li>\n<li>Identify the order of a differential equation.<\/li>\n<li>Find a solution to a linear, autonomous, first order differential equation.<\/li>\n<\/ul>\n<\/li>\n<li>Approximate solutions\n<ul>\n<li>Find equilibrium solutions to a differential equation.<\/li>\n<li>Use slope fields to sketch approximate solutions.<\/li>\n<li>Use Euler&#8217;s method to find approximate solutions.<\/li>\n<li>Construct a differential equation that would have a particular slope field \/ a given solution behavior.<\/li>\n<\/ul>\n<\/li>\n<li>Separable differential equations\n<ul>\n<li>Identify whether a differential equation is separable.<\/li>\n<li>Use separation of variables to solve separable differential equations.<\/li>\n<\/ul>\n<\/li>\n<li>Mass-spring systems\n<ul>\n<li>Find the characteristic equation for linear, constant coefficient, second order, homogeneous equations.<\/li>\n<li>Learn Euler&#8217;s formula.<\/li>\n<li>Use the characteristic equation to construct a general solution.<\/li>\n<li>Identify whether a differential equation has periodic \/ oscillatory solutions.<\/li>\n<li>Rewrite the second order equation as a system of two first order differential equations.<\/li>\n<\/ul>\n<\/li>\n<li>First order systems\n<ul>\n<li>Sketch a solution to a first order system in the phase plane.<\/li>\n<li>Distinguish between competition and predator-prey relationships in interaction equations.<\/li>\n<li>Draw nullclines in the phase plane.<\/li>\n<li>Do a phase plane analysis: identify all nullclines, find all equilibria, orient each nullcline, draw arrows indicating the flow direction within each subregion of the phase plane.<\/li>\n<li>Relate the values of dx\/dt and dy\/dt to the slope of the solution trajectory at a point.<\/li>\n<li>Sketch several representative solution trajectories in a phase plane, constructing a phase portrait.<\/li>\n<li>Work with the SIR model.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I have been using the Strogatz textbook for teaching dynamical systems. \u00a0The course has multivariable calculus and linear algebra prerequisites. \u00a0Students might take the prerequisite courses different places. \u00a0For students who have taken Math 1b, AM\/Math 21a, Math 21b, there was 6-7 week of differential equations background (11 classes in Math 1b + 9 classes [&hellip;]<\/p>\n","protected":false},"author":8032,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[157888,1010],"tags":[],"class_list":["post-108","post","type-post","status-publish","format-standard","hentry","category-dynamical-systems","category-math"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7E5LF-1K","jetpack-related-posts":[{"id":112,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-21b-differential-equations-background\/","url_meta":{"origin":108,"position":0},"title":"Dynamical Systems: Math 21b differential equations background","author":"siams","date":"10 June 2019","format":false,"excerpt":"For students who have taken Math 1b, AM\/Math 21a, Math 21b, there was 6-7 week of differential equations background (11 classes in Math 1b + 9 classes in 21b). \u00a0See my prior post for the Math 1b diff eq content that is relevant to Dynamical Systems. Student diff eq background\u2026","rel":"","context":"In &quot;Dynamical Systems&quot;","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":150,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/25\/varburg-and-purcell-7th-edition-differential-equations-mainly-chapter-18\/","url_meta":{"origin":108,"position":1},"title":"Varburg and Purcell 7th edition.  Differential equations (mainly chapter 18)","author":"siams","date":"25 June 2019","format":false,"excerpt":"Section 5.2: What is a diff eq?\u00a0 Provides an example and two solution methods before defining diff eq (and doesn't define a solution...).\u00a0 Then presents separation of variables via an example.\u00a0 Then a falling body example and an escape velocity example. Section 7.5: exponential growth and decay.\u00a0 They motivate y'\u2026","rel":"","context":"In &quot;Differential equations&quot;","block_context":{"text":"Differential equations","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/differential-equations\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":141,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/19\/hughes-hallett-et-al-chapter-11-differential-equations\/","url_meta":{"origin":108,"position":2},"title":"Hughes-Hallett et al Chapter 11: Differential equations","author":"siams","date":"19 June 2019","format":false,"excerpt":"11.1: What is a differential equation? Starts with an example: what sets the rate at which a person learns a new task? \u00a0Defines a diff eq and a solution to a diff eq. Defines order of a diff eq. \u00a0Example 1 is showing a function is not a solution to\u2026","rel":"","context":"In &quot;Math&quot;","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":146,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/25\/courant-and-john-1965-differential-equations-chapter-9\/","url_meta":{"origin":108,"position":3},"title":"Courant (and John) 1965, Differential Equations: Chapter 9.","author":"siams","date":"25 June 2019","format":false,"excerpt":"In the intro to Chapter 9 they note that we've already seen differential equations in Chapter 3, p. 223, and on p.312, and in Chapter 4 (see p 405).\u00a0 So I'll start there. Section 3.4: First encounter: in \"Some Applications of the Exponential Function\", y' = ay is introduced.\u00a0 \"Since\u2026","rel":"","context":"In &quot;Differential equations&quot;","block_context":{"text":"Differential equations","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/differential-equations-math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":153,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/28\/blanchard-devaney-and-hall-3rd-edition-2006-differential-equations\/","url_meta":{"origin":108,"position":4},"title":"Blanchard, Devaney, and Hall 3rd edition (2006): Differential Equations. Sections 1.1-1.4, 1.8","author":"siams","date":"28 June 2019","format":false,"excerpt":"Chapter 1: First order differential equations. \u00a0They present a goal: predicting a future value of a quantity modeled by a differential equation. Section 1.1a. \u00a0Modeling via differential equations. \u00a0a: Introduce the idea of a model. \u00a0Distinguish between the independent variable (time), dependent variables (dependent on time) and parameters (don't depend\u2026","rel":"","context":"In &quot;Differential equations&quot;","block_context":{"text":"Differential equations","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/differential-equations-math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":118,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-strogatz-chapter-2\/","url_meta":{"origin":108,"position":5},"title":"Dynamical Systems: Strogatz Chapter 2","author":"siams","date":"10 June 2019","format":false,"excerpt":"Following this text, students study 1d, then 2d, then 3d flows. \u00a0In 1d, we find stability, construct phase portraits, and in chapter 3, make bifurcation diagrams. \u00a0We loop back to these topics with more complexity in 2d. \u00a0This creates natural \"spacing\". A few notes on spacing: Spacing improves induction\/generalization from\u2026","rel":"","context":"In &quot;Dynamical Systems&quot;","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]}],"_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/108","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/users\/8032"}],"replies":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/comments?post=108"}],"version-history":[{"count":1,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/108\/revisions"}],"predecessor-version":[{"id":109,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/108\/revisions\/109"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=108"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=108"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=108"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}