{"id":681,"date":"2021-05-10T18:58:40","date_gmt":"2021-05-10T18:58:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=681"},"modified":"2021-11-17T02:43:52","modified_gmt":"2021-11-17T02:43:52","slug":"summary-of-basics-of-differential-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-basics-of-differential-equations\/","title":{"raw":"Summary of Basics of Differential Equations","rendered":"Summary of Basics of Differential Equations"},"content":{"raw":"<section id=\"fs-id1170571025566\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170571508562\" data-bullet-style=\"bullet\">\r\n \t<li>A differential equation is an equation involving a function [latex]y=f\\left(x\\right)[\/latex] and one or more of its derivatives. A solution is a function [latex]y=f\\left(x\\right)[\/latex] that satisfies the differential equation when [latex]f[\/latex] and its derivatives are substituted into the equation.<\/li>\r\n \t<li>The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.<\/li>\r\n \t<li>A differential equation coupled with an initial value is called an initial-value problem. To solve an initial-value problem, first find the general solution to the differential equation, then determine the value of the constant. Initial-value problems have many applications in science and engineering.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1170573740738\" class=\"section-exercises\" data-depth=\"1\"><\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170571278839\">\r\n \t<dt>differential equation<\/dt>\r\n \t<dd id=\"fs-id1170571278844\">an equation involving a function [latex]y=y\\left(x\\right)[\/latex] and one or more of its derivatives<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146732\">\r\n \t<dt>general solution (or family of solutions)<\/dt>\r\n \t<dd id=\"fs-id1170571146738\">the entire set of solutions to a given differential equation<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146742\">\r\n \t<dt>initial value(s)<\/dt>\r\n \t<dd id=\"fs-id1170571146747\">a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146753\">\r\n \t<dt>initial velocity<\/dt>\r\n \t<dd id=\"fs-id1170571146758\">the velocity at time [latex]t=0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146771\">\r\n \t<dt>initial-value problem<\/dt>\r\n \t<dd id=\"fs-id1170571146776\">a differential equation together with an initial value or values<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146780\">\r\n \t<dt>order of a differential equation<\/dt>\r\n \t<dd id=\"fs-id1170571146786\">the highest order of any derivative of the unknown function that appears in the equation<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146790\">\r\n \t<dt>particular solution<\/dt>\r\n \t<dd id=\"fs-id1170571146795\">member of a family of solutions to a differential equation that satisfies a particular initial condition<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571146801\">\r\n \t<dt>solution to a differential equation<\/dt>\r\n \t<dd id=\"fs-id1170571146806\">a function [latex]y=f\\left(x\\right)[\/latex] that satisfies a given differential equation<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1170571025566\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170571508562\" data-bullet-style=\"bullet\">\n<li>A differential equation is an equation involving a function [latex]y=f\\left(x\\right)[\/latex] and one or more of its derivatives. A solution is a function [latex]y=f\\left(x\\right)[\/latex] that satisfies the differential equation when [latex]f[\/latex] and its derivatives are substituted into the equation.<\/li>\n<li>The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.<\/li>\n<li>A differential equation coupled with an initial value is called an initial-value problem. To solve an initial-value problem, first find the general solution to the differential equation, then determine the value of the constant. Initial-value problems have many applications in science and engineering.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1170573740738\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170571278839\">\n<dt>differential equation<\/dt>\n<dd id=\"fs-id1170571278844\">an equation involving a function [latex]y=y\\left(x\\right)[\/latex] and one or more of its derivatives<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146732\">\n<dt>general solution (or family of solutions)<\/dt>\n<dd id=\"fs-id1170571146738\">the entire set of solutions to a given differential equation<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146742\">\n<dt>initial value(s)<\/dt>\n<dd id=\"fs-id1170571146747\">a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146753\">\n<dt>initial velocity<\/dt>\n<dd id=\"fs-id1170571146758\">the velocity at time [latex]t=0[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146771\">\n<dt>initial-value problem<\/dt>\n<dd id=\"fs-id1170571146776\">a differential equation together with an initial value or values<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146780\">\n<dt>order of a differential equation<\/dt>\n<dd id=\"fs-id1170571146786\">the highest order of any derivative of the unknown function that appears in the equation<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146790\">\n<dt>particular solution<\/dt>\n<dd id=\"fs-id1170571146795\">member of a family of solutions to a differential equation that satisfies a particular initial condition<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571146801\">\n<dt>solution to a differential equation<\/dt>\n<dd id=\"fs-id1170571146806\">a function [latex]y=f\\left(x\\right)[\/latex] that satisfies a given differential equation<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-681\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 2. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":416434,"menu_order":5,"template":"","meta":{"_candela_citation":"{\"1\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"}}","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-681","chapter","type-chapter","status-publish","hentry"],"part":159,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/681","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/416434"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/681\/revisions"}],"predecessor-version":[{"id":1036,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/681\/revisions\/1036"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/159"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/681\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=681"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=681"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=681"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=681"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}