[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571838-105":53,"doc-detail-571838-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","diff-eq-final","Diff Eq Final","","Final notes cover Newton’s Law of Cooling, methods for solving first-order linear differential equations, and approaches for mixing problems via concentration change reasoning. It explains second-order linear homogeneous equations using superposition and characteristic equations, then applies models to springs for free undamped, free damped, forced damped, and forced undamped (resonance) motion, distinguishing transient and steady-state behavior. It also introduces Laplace transforms, linearity and shifting theorems, inverse and eigenvalue-based solution methods for linear systems.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/diff-eq-final/571838/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/diff-eq-final/571838.png","ImageObject",442,249,{"name":88,"@type":89},"8796093062539","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-20","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How is a first-order linear differential equation solved using an integrating factor?","Question",{"text":108,"@type":109},"Write the equation in standard form, divide by the coefficient of y (or the required term) to match the standard pattern, then multiply both sides by the integrating factor. Reverse the product rule and integrate both sides with respect to x.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How does the solution form of a second-order homogeneous equation depend on the characteristic roots?",{"text":113,"@type":109},"Use the characteristic equation and classify roots: real distinct roots lead to two independent real solutions; complex roots produce sinusoidal forms; repeated real roots give solutions that include a polynomial factor times an exponential.",{"name":115,"@type":106,"acceptedAnswer":116},"What key properties are used for the Laplace transform?",{"text":117,"@type":109},"The notes define the Laplace transform, use linearity to split sums, and apply shifting theorems to handle multiplied-by-exponential and time-shifted forms. These properties support solving differential equations in the transform domain and then converting back via inverses.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},571838,1792524703,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":60,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":129,"file_id":130,"file_url":131,"file_type":132,"file_size":133,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":73,"language":134,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":135,"faqs":136,"seo_title":137,"seo_description":61,"update_tm":138,"read_time":9},8796093062539,"| Newton’s Law of Cooling\u003Cbr>􀀡􀀡􀀢􀀣 = −􀀧 (􀀢 − 􀀩)\u003Cbr>Where A is the temperature of the surrounding environment, andk is the constant of proportionality.\u003Cbr>􀀢 = 􀀫􀀬 -./ + 􀀩\u003Cbr>Solving First Order Linear Eqs.\u003Cbr>- Given an FODE of the form\u003Cbr>􀀱2 (􀀳)􀀴 5 + 􀀱6(􀀳)􀀴 = 􀀷 (􀀳)\u003Cbr>Divide both sides by 􀀱2 (􀀳) to get an equation of the standard form\u003Cbr>􀀴 5 + 􀀸 (􀀳)􀀴 = 􀀹 (􀀳)\u003Cbr>Then multiply both sides by the integrating factor 􀀺 (􀀳) = 􀀬 ∫ \u003C(=)>=\u003Cbr>To reverse product rule and then integrate both sides with respect to x.\u003Cbr>Mixing Problems\u003Cbr>􀀡􀀡􀀩􀀣 = ?􀁀􀁁􀁂􀀬􀁂􀁄􀀣􀁃􀁂􀀱􀀣􀁄􀁁􀁂 ∗ 􀁃􀀱􀀣􀁄􀁂􀀬 F 􀁀􀁁􀁂􀁀􀀬􀁂􀀣􀁃􀀱􀀣􀁄􀁁􀁂 􀁃􀀱􀀣􀀬\u003Cbr>−( ∗ )\u003Cbr>􀁁􀁇􀀣 􀁁􀁇􀀣\u003Cbr>Note: The net volume might be changing so make sure to reflect that in the concentration out field.\u003Cbr>Second Order Linear Eqs.\u003Cbr>- Given a homogenous SODE of the form\u003Cbr>􀀱6 (􀀳)􀀴 55 + 􀀱2(􀀳)􀀴 5 + 􀀱6 (􀀳)􀀴 = 0\u003Cbr>The principle of superposition states that if 􀀴2(􀀳) and\u003Cbr>􀀴6(􀀳) are linearly independent solutions, then 􀀴 =\u003Cbr>􀁀2􀀴2 (􀀳) + 􀁀6􀀴6 (􀀳) is also a solution.\u003Cbr>- Given a homogenous SODE of Constant Coefficients 􀀱􀀴 55 + 􀁉􀀴 5 + 􀁀􀀴 = 0\u003Cbr>Its characteristic equation is given by\u003Cbr>􀀱􀁃6 + 􀁉􀁃 + 􀁀 = 0\u003Cbr>And solving for r can help us determine the form of the solution.\u003Cbr>−􀁉 ± √􀁉 6 − 4􀀱􀁀􀁃 = ~~ ~~\u003Cbr>2􀀱\u003Cbr>o Case 1: 􀁉 6 − 4􀀱􀁀 > 0 à roots are real and distinct\u003Cbr>§ General Solution: 􀀴 = 􀁀2􀀬 OP= + 􀁀6􀀬 OQ=\u003Cbr>o Case 2: 􀁉6 − 4􀀱􀁀 \u003C 0 à roots complex\u003Cbr>􀁃 = 􀁓 ± 􀁔􀁄\u003Cbr>§ General Solution: 􀀴 = 􀀬 U= (􀁀2 cos(􀁔􀀳) +\u003Cbr>􀁀6 sin(􀁔􀀳))\u003Cbr>o Case 1: 􀁉6 − 4􀀱 => 0 à roots are real and equal 􀁉\u003Cbr>􀁃 = − ~~  ~~2􀀱\u003Cbr>§ General Solution: 􀀴 = 􀁀2􀀬 O= + 􀁀6􀀳􀀬 O=\u003Cbr>Springs\u003Cbr>- Note: Displacements are positive in the downward direction and negative in the upward direction.\u003Cbr>- The motion of a spring is modeled by the diff eq.\u003Cbr>􀁛􀀳 55 + 􀁀􀀳 5 + 􀀧􀀳 = 􀀷 (􀀣) Where m is the mass attached, c ≥ 0 is the damping constant, and k is the spring constant.\u003Cbr>o Case 1: Free Undamped Motion (f(t)=0, c=0)\u003Cbr>§ 􀁛􀀳 55 + 􀀧􀀳 = 0 → 􀀳 55 + ].~~ ~~ 􀀳 = 0\u003Cbr>§ 􀁞 = _]. (circular frequency)\u003Cbr>§ 􀀳 (􀀣) = 􀁀2 cos(􀁞􀀣) + 􀁀6sin (􀁞􀀣)\u003Cbr>§ 􀀩 = `􀁀62 + 􀁀66\u003Cbr>• cos(􀁡) = b~~ ~~Pc\u003Cbr>• sin(􀁡) = b~~ ~~Qc\u003Cbr>§ Phase angle 􀁡 :\u003Cbr>• 1st quadrant à 􀁡 = arctan (b~~ ~~Pb~~ ~~Q)\u003Cbr>• 2nd quadrant à 􀁡 = 􀁧 + arctan (b~~ ~~Pb~~ ~~Q)\u003Cbr>• 3rd quadrant à 􀁡 = 􀁧 + arctan (b~~ ~~Pb~~ ~~Q)\u003Cbr>• 4nd quadrant à 􀁡 = 2􀁧 + arctan (b~~ ~~Pb~~ ~~Q)\u003Cbr>§ Amplitude-phase form:\u003Cbr>• 􀀳 (􀀣) = 􀀩􀁀􀁁􀁨 (􀁞􀀣 − 􀁡)\u003Cbr>o Case 2: Free Damped Motion (f(t)=0, c>0)\u003Cbr>§ 􀁛􀀳 55 + 􀁀􀀳 5 + 􀀧􀀳 = 0\u003Cbr>• Characteristic eq: 􀁛􀁃6 + 􀁀􀁃 + 􀀧 = 0\u003Cbr>• 􀁃 = -b±√b6Q]-i].\u003Cbr>§ 􀁀6 − 4􀁛􀀧 > 0 à 2 real distinct roots\u003Cbr>• Motion is overdamped\u003Cbr>§ 􀁀6 − 4􀁛􀀧 = 0 à 2 real equal roots\u003Cbr>• Motion is critically damped\u003Cbr>§ 􀁀6 − 4􀁛􀀧 \u003C 0 à 2 complex roots\u003Cbr>• Motion is underdamped\u003Cbr>• 􀁃 = 􀁪 + 􀁫􀁄\u003Cbr>• 􀀳 (􀀣) = 􀀬 l/ (􀁀2 cos(􀁫􀀣) +\u003Cbr>􀁀6 sin(􀁫􀀣))\u003Cbr>In amplitude-phase form\u003Cbr>• 􀀳 (􀀣) = 􀀩􀀬l/ cos(􀁫􀀣 − 􀁡)\u003Cbr>Where 􀁫 is the pseudo-frequnecy. | Nonhomogeneous Equations and Undetermined Coefficients\u003Cbr>- The solution to the nonhomogeneous equation of the form 􀀱􀀴 55 + 􀁉􀀴 5 + 􀁀􀀴 = 􀀷 (􀀳)\u003Cbr>Is given by 􀀴 = 􀀴b + 􀀴l , where 􀀴b (called the complimentary solution) is the general solution to 􀀱􀀴 55 + 􀁉􀀴 5 + 􀁀􀀴 = 0, and\u003Cbr>􀀴l (called the particular solution is any function satisfying 􀀱􀀴 55 + 􀁉􀀴 5 + 􀁀􀀴 = 􀀷 (􀀳) .\u003Cbr>- 􀀴l can be found using the method of undetermined coefficients:\u003Cbr>If 􀀱􀀴 55 + 􀁉􀀴 5 + 􀁀􀀴 = 􀀷 (􀀳) Then the particular solution 􀀴lis 1. f(x) = a polynomial in x 􀀴l = 􀀳.(a general polynomial of the same degree)\u003Cbr>- where k is the number of times that 0 is a root of the characteristic equation 2. 􀀷 (􀀳) = 􀀬 m= (a polynomial in x) 􀀴l = 􀀳.􀀬 m=(a general polynomial of the same degree)\u003Cbr>- where k is the number of times that a is a root of the characteristic equation 3. 􀀷 (􀀳) = 􀀬 m=cos (􀁉􀀳)\u003Cbr>(a polynomial in x)\u003Cbr>Or\u003Cbr>4. 􀀷 (􀀳) = 􀀬 m=cos (􀁉􀀳)\u003Cbr>(a polynomial in x) 􀀴l = 􀀳.􀀬 m= [(polynomial of same degree)cos(􀁉􀀳) +(another polynomial of same degree)sin(􀁉􀀳)]\u003Cbr>- where k is the number of times that 􀀱 ± 􀁉􀁄 are the roots of the characteristic equation.\u003Cbr>More Springs\u003Cbr>- Case 3: Forced D","cbCaicJQ33ABy4NO","https://ap.wps.com/l/cbCaicJQ33ABy4NO","pdf",506999,"English","# Solving Differential Equations\n## Newton’s Law of Cooling\n## Solving First Order Linear Equations\n## Mixing Problems\n## Second Order Linear Equations\n## Springs Modeling\n## Nonhomogeneous Equations and Undetermined Coefficients\n## Forced Motion and Resonance\n## Laplace Transform\n## Inverses and Eigenvectors\n## Eigenvalue Method for Linear Systems","[{\"question\":\"How is a first-order linear differential equation solved using an integrating factor?\",\"answer\":\"Write the equation in standard form, divide by the coefficient of y (or the required term) to match the standard pattern, then multiply both sides by the integrating factor. Reverse the product rule and integrate both sides with respect to x.\"},{\"question\":\"How does the solution form of a second-order homogeneous equation depend on the characteristic roots?\",\"answer\":\"Use the characteristic equation and classify roots: real distinct roots lead to two independent real solutions; complex roots produce sinusoidal forms; repeated real roots give solutions that include a polynomial factor times an exponential.\"},{\"question\":\"What key properties are used for the Laplace transform?\",\"answer\":\"The notes define the Laplace transform, use linearity to split sums, and apply shifting theorems to handle multiplied-by-exponential and time-shifted forms. These properties support solving differential equations in the transform domain and then converting back via inverses.\"}]","Diff Eq Final | PDF",1791923991]