{"id":150,"date":"2019-06-25T17:55:58","date_gmt":"2019-06-25T21:55:58","guid":{"rendered":"https:\/\/blogs.harvard.edu\/siams\/?p=150"},"modified":"2019-06-28T09:53:21","modified_gmt":"2019-06-28T13:53:21","slug":"varburg-and-purcell-7th-edition-differential-equations-mainly-chapter-18","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/25\/varburg-and-purcell-7th-edition-differential-equations-mainly-chapter-18\/","title":{"rendered":"Varburg and Purcell 7th edition.  Differential equations (mainly chapter 18)"},"content":{"rendered":"<ul>\n<li>Section 5.2: What is a diff eq?\u00a0 Provides an example and two solution methods before defining diff eq (and doesn&#8217;t define a solution&#8230;).\u00a0 Then presents separation of variables via an example.\u00a0 Then a falling body example and an escape velocity example.<\/li>\n<li>Section 7.5: exponential growth and decay.\u00a0 They motivate y&#8217; = ky via population growth, then solve by separation.\u00a0 Example 1: doubling time.\u00a0 Example 2: growth time.\u00a0 Example 3: radioactive decay.\u00a0 Example 4 and 5: compound interest.<\/li>\n<li>Extra note: in section 3.10 they present little-o notation.<\/li>\n<li>Chapter 18: differential equations.\u00a0 Section 18.1: linear first order equations.\u00a0 They define &#8220;differential equations&#8221;, &#8220;ordinary differential equation of order n&#8221;, &#8220;solution&#8221;, &#8220;general solution&#8221;, &#8220;particular solution&#8221;, &#8220;linear&#8221;.\u00a0 They introduce the method of integrating factors, and apply to a mixture problem, to a circuit, and to a battery.<\/li>\n<li>Section 18.2: second order homogeneous equations.\u00a0 They define &#8220;independent&#8221; for solutions, the auxiliary equation, and use it to provide a solution to a diff eq with constant coefficients.\u00a0 They don&#8217;t intro Euler&#8217;s formula but assume it in the complex roots example.\u00a0 Then on to higher order equations.<\/li>\n<li>Section 18.3: the nonhomogeneous equation.\u00a0 they provide general \/ particular solution info, do the method of undetermined coefficients, and variation of parameters.<\/li>\n<li>Section 18.4: Applications of second order equations.\u00a0 A vibrating spring, simple harmonic motion, damping, overdamped, critically damped.\u00a0 \u00a0Electric circuits.<\/li>\n<li>This text goes through a laundry-list of methods.\u00a0 Perhaps not so much motivation.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Section 5.2: What is a diff eq?\u00a0 Provides an example and two solution methods before defining diff eq (and doesn&#8217;t define a solution&#8230;).\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&#8217; = ky via population growth, [&hellip;]<\/p>\n","protected":false},"author":8032,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_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},"jetpack_post_was_ever_published":false},"categories":[157889,1010],"tags":[],"class_list":["post-150","post","type-post","status-publish","format-standard","hentry","category-differential-equations","category-math"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7E5LF-2q","jetpack-related-posts":[{"id":141,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/19\/hughes-hallett-et-al-chapter-11-differential-equations\/","url_meta":{"origin":150,"position":0},"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":153,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/28\/blanchard-devaney-and-hall-3rd-edition-2006-differential-equations\/","url_meta":{"origin":150,"position":1},"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":146,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/25\/courant-and-john-1965-differential-equations-chapter-9\/","url_meta":{"origin":150,"position":2},"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":108,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-1b-differential-equations-background\/","url_meta":{"origin":150,"position":3},"title":"Dynamical systems: Math 1b differential equations background.","author":"siams","date":"10 June 2019","format":false,"excerpt":"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\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":112,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-21b-differential-equations-background\/","url_meta":{"origin":150,"position":4},"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":118,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-strogatz-chapter-2\/","url_meta":{"origin":150,"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\/150","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=150"}],"version-history":[{"count":1,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/150\/revisions"}],"predecessor-version":[{"id":151,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/150\/revisions\/151"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=150"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=150"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=150"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}