{"id":153,"date":"2019-06-28T10:04:48","date_gmt":"2019-06-28T14:04:48","guid":{"rendered":"https:\/\/blogs.harvard.edu\/siams\/?p=153"},"modified":"2019-06-28T10:19:37","modified_gmt":"2019-06-28T14:19:37","slug":"blanchard-devaney-and-hall-3rd-edition-2006-differential-equations","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/28\/blanchard-devaney-and-hall-3rd-edition-2006-differential-equations\/","title":{"rendered":"Blanchard, Devaney, and Hall 3rd edition (2006): Differential Equations. Sections 1.1-1.4, 1.8"},"content":{"rendered":"<p>Chapter 1: First order differential equations. \u00a0They present a goal: predicting a future value of a quantity modeled by a differential equation.<\/p>\n<ul>\n<li>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&#8217;t depend on time but can be adjusted).<\/li>\n<li>Section 1.1b. \u00a0Modeling via differential equations. \u00a0b: Unlimited population growth. \u00a0P&#8217; = k P is the equation (exponential growth). \u00a0\u00a0Define first-order, ordinary differential equation, equilibrium solution, initial condition, qualitative analysis. \u00a0Introduce initial-value-problem, and solution. \u00a0Guess and check method of finding a solution. \u00a0Particular solution vs general solution. \u00a0Example comparing to United States population (annual census since 1790).<\/li>\n<li>Section 1.1c. \u00a0Modeling via differential equations. \u00a0c: Logistic population growth. \u00a0Add a second assumption (at some level of population growth will become negative). \u00a0Logistic population model, nonlinear, equilibria. \u00a0They do a qualitative analysis and create approximate solutions.<\/li>\n<li>Section 1.1d. \u00a0Modeling via differential equations. \u00a0d: Predator prey systems. \u00a0Add assumptions about fox and rabbit interactions. \u00a0They generate a first order system and define the solution to a system.<\/li>\n<li>Section 1.1e. \u00a0Modeling via differential equations. \u00a0e: Analytic, qualitative, and numerical approaches. \u00a0Here they name that there are three approaches.<\/li>\n<li>Section 1.2a. \u00a0Analytic technique: separation of variables. \u00a0a. \u00a0What is a differential equation and what is a solution?<\/li>\n<li>Section 1.2b. \u00a0Analytic technique: separation of variables. \u00a0b. \u00a0Initial-value problems and the general solution.<\/li>\n<li>Section 1.2c. \u00a0Analytic technique: separation of variables. \u00a0c. \u00a0Initial-value problems and the general solution.<\/li>\n<li>Section 1.2d. \u00a0Analytic technique: separation of variables. \u00a0d. \u00a0Separable equations<\/li>\n<li>Section 1.2e. \u00a0Analytic technique: separation of variables. \u00a0e. \u00a0Missing solutions<\/li>\n<li>Section 1.2f. \u00a0Analytic technique: separation of variables. \u00a0f. Getting stuck<\/li>\n<li>Section 1.2g. \u00a0Analytic technique: separation of variables. \u00a0g. A savings model<\/li>\n<li>Section 1.2h. \u00a0Analytic technique: separation of variables. \u00a0h. A mixing problem<\/li>\n<li>Section 1.3a. \u00a0Qualitative technique: slope fields. \u00a0a. The geometry of dy\/dt = f(t,y)<\/li>\n<li>Section 1.3b. \u00a0Qualitative technique: slope fields. \u00a0b. Slope fields<\/li>\n<li>Section 1.3c. \u00a0Qualitative technique: slope fields. \u00a0c. Important special cases<\/li>\n<li>Section 1.3d. \u00a0Qualitative technique: slope fields. \u00a0d. Analytic versus qualitative analysis<\/li>\n<li>Section 1.3e. \u00a0Qualitative technique: slope fields. \u00a0e. The mixing problem revisited<\/li>\n<li>Section 1.3f. \u00a0Qualitative technique: slope fields. \u00a0f. An RC circuit<\/li>\n<li>Section 1.3g. \u00a0Qualitative technique: slope fields. \u00a0g. Combining qualitative with quantitative results<\/li>\n<li>Section 1.4a. \u00a0Numerical technique: Euler&#8217;s method. \u00a0a. Stepping along the slope field<\/li>\n<li>Section 1.4b. \u00a0Numerical technique: Euler&#8217;s method. \u00a0b. Euler&#8217;s method<\/li>\n<li>Section 1.4c. \u00a0Numerical technique: Euler&#8217;s method. \u00a0c. Approximating an autonomous equation<\/li>\n<li>Section 1.4d. \u00a0Numerical technique: Euler&#8217;s method. \u00a0d. A non-autonomous example<\/li>\n<li>Section 1.4e. \u00a0Numerical technique: Euler&#8217;s method. \u00a0e. An RC circuit with periodic input<\/li>\n<li>Section 1.4f. \u00a0Numerical technique: Euler&#8217;s method. \u00a0f. Errors in numerical methods<\/li>\n<li>Leave existence and uniqueness (1.5), equilibria and the phase line (1.6), bifurcations (1.7), integrating factors (1.9) to a later course.<\/li>\n<li>Section 1.8a. \u00a0Linear equations. \u00a0a. \u00a0Linear differential equations<\/li>\n<li>Section 1.8b. \u00a0Linear equations. \u00a0b. \u00a0Linearity principles<\/li>\n<li>Section 1.8c. \u00a0Linear equations. \u00a0c. \u00a0Solving linear equations<\/li>\n<li>Section 1.8d. \u00a0Linear equations. \u00a0d. \u00a0Qualitative analysis<\/li>\n<li>Section 1.8e. \u00a0Linear equations. \u00a0e. \u00a0Second guessing<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>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&#8217;t depend on time but can be [&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_feature_clip_id":0,"_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":[157890,1010],"tags":[],"class_list":["post-153","post","type-post","status-publish","format-standard","hentry","category-differential-equations-math","category-math"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7E5LF-2t","jetpack-related-posts":[{"id":146,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/25\/courant-and-john-1965-differential-equations-chapter-9\/","url_meta":{"origin":153,"position":0},"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":150,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/25\/varburg-and-purcell-7th-edition-differential-equations-mainly-chapter-18\/","url_meta":{"origin":153,"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":153,"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":108,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-1b-differential-equations-background\/","url_meta":{"origin":153,"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":153,"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":170,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/10\/meiss-differential-dynamical-systems-chaos\/","url_meta":{"origin":153,"position":5},"title":"Meiss: Differential Dynamical Systems (chaos)","author":"siams","date":"10 July 2019","format":false,"excerpt":"I am reading James Meiss' text Differential Dynamical Systems (SIAM). \u00a0I am specifically interested in how he tells the story of chaos. In the Preface, he mentions the following: That\u00a0\u00a0Chapter 5 focuses on invariant manifolds: stable and unstable sets heteroclinic orbits stable manifolds local stable manifold theorem global stable manifolds\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\/153","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=153"}],"version-history":[{"count":5,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/153\/revisions"}],"predecessor-version":[{"id":158,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/153\/revisions\/158"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=153"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=153"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=153"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}