{"id":51,"date":"2004-12-05T14:59:11","date_gmt":"2004-12-05T14:59:11","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=51"},"modified":"2004-12-05T14:59:11","modified_gmt":"2004-12-05T14:59:11","slug":"agnews-differential-equations","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=51","title":{"rendered":"Agnew&#8217;s Differential Equations"},"content":{"rendered":"<p>When I studied differential equations, the set book was Ralph P Agnew&#8217;s <a href=\"http:\/\/www.amazon.co.uk\/exec\/obidos\/ASIN\/0070005990\/qid=1102257690\/sr=1-1\/ref=sr_1_0_1\/202-3869318-0022214\" target=\"_blank\">Differential Equations<\/a>. It had a brilliant index which seemed to contain every word in the book. Everything I ever wanted to look up was referenced in that index, unlike plenty of other textbooks.<\/p>\n<p>It included a wonderful polemic about textbooks that claim that differential equations of order n have a general solution with n essential (aka arbitrary) constants: <\/p>\n<ul><i>The promoted the view that to each differential equation of order n there corresponds an important family of solutions from which all other solutions &#8230; are obtainable by use of appropriate hocus-pocus involving envelopes and more complicated things. It was essential that this family of solutions should have a name (this is the start of the intimidation) which would immediately convince everybody that it existed and was important. With dubious regard for appropriateness of terminology, this family was called &#8220;the general solution&#8221; of the given equation.<\/i><\/ul>\n<p>He then says: <\/p>\n<ul><i>It may be unclear whether this <\/i>[differential equations of order n have n essential constants] <i>is a theorem or a definition or merely a collection of words, but we are now in a realm where nearly everything is unclear. One thing, however, is clear. No meaning has been attached to the statement that a formula has n eseential constant. This gives the good old lecturer a chance to practice the art of proof by intimidation.<\/i><\/ul>\n<p>He then goes on to justify his remarks and includes as examples the differential equations <img src='\/maths\/latexrender\/pictures\/a5d66d579f5522e380d2a066af37911c.gif' title='\\displaystyle \\left|\\frac{dy}{dx}\\right|+|x|+|y|+1=0' alt='\\displaystyle \\left|\\frac{dy}{dx}\\right|+|x|+|y|+1=0' align=absmiddle> which has no solutions and <img src='\/maths\/latexrender\/pictures\/7f173e998bed9b42b6d79fec2bb53996.gif' title='\\displaystyle \\left(\\frac{dy}{dx}\\right)^2+y^2=1' alt='\\displaystyle \\left(\\frac{dy}{dx}\\right)^2+y^2=1' align=absmiddle> which has &#8220;<i>vast hordes of real solutions<\/i>&#8220;.<\/p>\n<p>He also deals with the solution of <img src='\/maths\/latexrender\/pictures\/8c48052bff77455a254de8ab16ebfa4c.gif' title='\\displaystyle \\frac{dy}{dx}=ky' alt='\\displaystyle \\frac{dy}{dx}=ky' align=absmiddle> where dividing by <img src='\/maths\/latexrender\/pictures\/415290769594460e2e485922904f345d.gif' title='y' alt='y' align=absmiddle> (the usual method taught at A level) won&#8217;t do since it could involve division by zero (the book indexes this as <i>Division by zero taboo<\/i>).<\/p>\n<p>Finally, there is the wonderful snowplough problem (or <i>snowplow<\/i> as the author is American) which says: <\/p>\n<ul><i>One day it started snowing at a heavy and steady rate. A snowplow started out at noon, going 2 miles the first hour and 1 mile the second hour. What time did it start snowing?<\/i><\/ul>\n<p>He says &#8220;<i>Our first task is to recover from the shock of being asked to solve such a problem&#8221;<\/i> and goes on &#8220;<i>we assume that the plow clears snow at a constant rate of k cubic miles per hour<\/i>&#8220;.<\/p>\n<p>Now it&#8217;s up to you. Click on <b>read more<\/b> below for the time it started snowing.<br \/>\n<!--more--><br \/>\nIt started snowing at around 11:23am. More precisely, <img src='\/maths\/latexrender\/pictures\/dbccaf7cf2a0bdc90c26dd86d2073092.gif' title='\\frac{-1+\\sqrt{5}}{2}' alt='\\frac{-1+\\sqrt{5}}{2}' align=absmiddle> hours before midday.<br \/>\nIf you&#8217;re desperate the solution is in <a href=\"http:\/\/www.as.ysu.edu\/~faires\/Courses\/2002-2003\/03M3705\/snowplow.pdf\" target=\"_blank\">The Snowplow Problem by Ralph Agnew<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>When I studied differential equations, the set book was Ralph P Agnew&#8217;s Differential Equations. It had a brilliant index which seemed to contain every word in the book. Everything I ever wanted to look up was referenced in that index, unlike plenty of other textbooks. It included a wonderful polemic about textbooks that claim that [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-51","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/51","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=51"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/51\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=51"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=51"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=51"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}