{"id":146,"date":"2019-06-25T17:32:24","date_gmt":"2019-06-25T21:32:24","guid":{"rendered":"https:\/\/blogs.harvard.edu\/siams\/?p=146"},"modified":"2019-06-28T09:52:46","modified_gmt":"2019-06-28T13:52:46","slug":"courant-and-john-1965-differential-equations-chapter-9","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/25\/courant-and-john-1965-differential-equations-chapter-9\/","title":{"rendered":"Courant (and John) 1965, Differential Equations: Chapter 9."},"content":{"rendered":"<p>In the intro to Chapter 9 they note that we&#8217;ve already seen differential equations in Chapter 3, p. 223, and on p.312, and in Chapter 4 (see p 405).\u00a0 So I&#8217;ll start there.<\/p>\n<ul>\n<li>Section 3.4: First encounter: in &#8220;Some Applications of the Exponential Function&#8221;, y&#8217; = ay is introduced.\u00a0 &#8220;Since Eq. (8) expresses a relation between the function and its derivative, it is called the differential equation of the exponential function&#8221;.\u00a0 They show that the exponential function is the unique solution (this argument is worthwhile, actually, because it is a small proof).<\/li>\n<li>Section 3.4: more y&#8217; = ay.\u00a0 Examples associated with the exponential function: compound interest, radioactive decay, Newton&#8217;s law of cooling, atmospheric pressure with height above the surface of the Earth,\u00a0 the law of mass action (chemical reactions), switching on and off an electric circuit.\u00a0 Newton&#8217;s law of cooling,\u00a0 the law of mass action, and the electric circuit involve differential equations.<\/li>\n<li>Section 3.16a: Differential equations of trigonometric functions.\u00a0 In 3.16a they intro diff eqs.\u00a0 Diff eqs move beyond equations y&#8217; = f(x) to &#8220;more general relationships between y and derivatives of y&#8221;.<\/li>\n<li>Section 3.16b: Define sine and cosine via a differential equation (u&#8221; + u = 0) and an initial condition.\u00a0 &#8220;Any function u = F(x) satisfying the equation, &#8230;, is a solution.&#8221;\u00a0 They then show that shifts of solutions are solutions and linear combos are solutions, and scalar multiples are solutions, so the properties of linearity.\u00a0 Initial conditions single out a specific solution.\u00a0 They also derive cos(x+y) = cos x cos y &#8211; sin x sin y using the differential equation.\u00a0 They also note that pi\/2 can then be defined via &#8220;the smallest positive value of x for which cos x = 0.&#8221;<\/li>\n<li>Section 4.4a: Newton&#8217;s law of motion, a relationship &#8220;from which we hope to determine the motion&#8221;.\u00a0 They define diff eq and solution again.<\/li>\n<li>Section 4.4b\/c: Motion of falling bodies and motion constrained to a curve.<\/li>\n<li>Section 4.5: free fall of a body in the air (find terminal velocity under this model)<\/li>\n<li>Section 4.6: simplest elastic vibration: motion of a spring.<\/li>\n<li>Section 4.7abcde: motion on a given curve.\u00a0 The differential equation and its solution.\u00a0 Particle sliding down a curve.\u00a0 Discussion of the motion.\u00a0 The ordinary pendulum.\u00a0 The cycloidal pendulum.<\/li>\n<li>Section 4.8abc: Motion in a gravitational field.\u00a0 Newton&#8217;s universal law of gravitation.\u00a0 Circular motion about the center of attraction.\u00a0 Radial motion &#8211; escape velocity.<\/li>\n<li>Chapter 9:\u00a0 They summarize the differential differential equations that have been encountered above.\u00a0 This chapter is differential equations for the simplest types of vibration.<\/li>\n<li>9.1ab: Vibration problems of mechanics and physics.\u00a0 The simplest mechanical vibrations (forced second order constant coefficient equation).\u00a0 Electrical oscillations (similar).<\/li>\n<li>9.2abc: Solution of the homogeneous equation.\u00a0 Free oscillations.\u00a0a:The formal solution.\u00a0 b:Interpretation of the solution.\u00a0 c: Fulfilment of given initial conditions. Uniqueness of the solution. In the formal solution they construct the characteristic equation and distinguish the three cases of roots.\u00a0 For the solutions in complex form they introduce Euler&#8217;s formula.\u00a0 In interpretation they introduce &#8220;damping&#8221;, &#8220;damped harmonic oscillations&#8221;, &#8220;attenuation constant&#8221;, &#8220;natural frequency&#8221;.<\/li>\n<li>9.3abcde: The nonhomogeneous equation.\u00a0 Forced oscillations.\u00a0 a: general remards.\u00a0 Superposition.\u00a0 b: Solution of the nonhomogeneous equation.\u00a0 c: The resonance curve.\u00a0 d: Further discussion of the oscillation.\u00a0 e: Remarks on the construction of recording instruments.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>In the intro to Chapter 9 they note that we&#8217;ve already seen differential equations in Chapter 3, p. 223, and on p.312, and in Chapter 4 (see p 405).\u00a0 So I&#8217;ll start there. Section 3.4: First encounter: in &#8220;Some Applications of the Exponential Function&#8221;, y&#8217; = ay is introduced.\u00a0 &#8220;Since Eq. (8) expresses a relation [&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":[157890,1010],"tags":[],"class_list":["post-146","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-2m","jetpack-related-posts":[{"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":146,"position":0},"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":153,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/28\/blanchard-devaney-and-hall-3rd-edition-2006-differential-equations\/","url_meta":{"origin":146,"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":141,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/19\/hughes-hallett-et-al-chapter-11-differential-equations\/","url_meta":{"origin":146,"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":112,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-21b-differential-equations-background\/","url_meta":{"origin":146,"position":3},"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":108,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-1b-differential-equations-background\/","url_meta":{"origin":146,"position":4},"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":170,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/10\/meiss-differential-dynamical-systems-chaos\/","url_meta":{"origin":146,"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\/146","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=146"}],"version-history":[{"count":3,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/146\/revisions"}],"predecessor-version":[{"id":149,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/146\/revisions\/149"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=146"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=146"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=146"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}