[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571841-105":53,"doc-detail-571841-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","ode-cheat-sheet-first-order-equations-solution-methods","ODE Cheat Sheet - First Order Equations - Solution Methods","","Concise cheat-sheet covering ordinary differential equations with step-by-step solution frameworks. Topics include first-order linear equations using integrating factors, separable equations, exact equations with the condition My=Nx, and transformations for non-exact forms. Extends to second-order linear ODEs with constant and Cauchy-Euler coefficients, nonhomogeneous methods (undetermined coefficients, reduction of order, variation of parameters), and applications such as free fall, population dynamics, and Newton’s law of cooling. Also summarizes numerical schemes, series solutions, singular points with Frobenius method, Laplace transforms, special functions, and systems via eigenvalues and matrix solutions.",{"@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":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/ode-cheat-sheet-first-order-equations-solution-methods/571841/",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/ode-cheat-sheet-first-order-equations-solution-methods/571841.png","ImageObject",442,249,{"name":88,"@type":89},"Jiven","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-24","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":47},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How do you solve first-order linear differential equations using an integrating factor?","Question",{"text":108,"@type":109},"Rewrite the equation as y'(x)+p(x)y(x)=f(x), compute the integrating factor μ(x)=exp(∫p(x)dx), then use (μy)'=μf to integrate and add the constant C.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What is the condition for an equation to be exact in the form M(x,y)dx+N(x,y)dy=0?",{"text":113,"@type":109},"Exactness requires My(x,y)=Nx(x,y). When this holds, introduce a potential u(x,y) so that du=Mdx+Ndy and set u(x,y)=const.",{"name":115,"@type":106,"acceptedAnswer":116},"Which method is used for solving second-order nonhomogeneous ODEs and how is it selected?",{"text":117,"@type":109},"Use the method of undetermined coefficients when the forcing term matches standard guess patterns. For reduction of order, start from a known homogeneous solution y1. Use variation of parameters when writing the particular solution as y_p=c1(x)y1(x)+c2(x)y2(x) with auxiliary equations for c1' and c2'.","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},571841,1792048156,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":47,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":73,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":9},1099513958607,"https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399","ODE Cheat Sheet  \nFirst Order Equations  \nSeparable  \ny0 (x) = f (x)g (y)  \nR gd(yy) = R f (x) dx + C  \nLinear First Order  \ny0 (x) + p(x)y (x) = f (x)  \n􀀖 (x) = exp Rx p (􀀘) d􀀘 Integrating factor.  \n(􀀖y)0 = f 􀀖 Exact Derivative.  \nSolution: y (x) = 􀀖~~ 1~~(x) 􀀀R f(􀀘)􀀖 (􀀘) d􀀘 + C􀀁  \nExact  \n0 = M (x; y)dx + N(x; y)dy Solution: u (x; y) = const where  \ndu = ~~@~~@~~u~~x dx + ~~@~~@~~u~~y dy Condition: My = Nx  \n~~@~~@~~u~~x = M (x; y) ; ~~ @~~@~~u~~y = N (x; y)  \nNon-Exact Form  \n􀀖 (x; y) (M(x; y) dx + N(x; y) dy) = du (x; y) My = Nx  \nN @@􀀖x 􀀀 M @@􀀖y = 􀀖 􀀀 ~~@~~@My 􀀀 ~~@~~@Nx 􀀁 :  \nSpecial cases  \nIf MyNx = h (y) ; then 􀀖(y) = exp R h (y) dy  \nIf MyNx = 􀀀h(x) ; then 􀀖(y) = exp R h (x) dx  \nSecond Order Equations  \nLinear  \na (x)y00 (x) + b(x)y0 (x) + c(x)y (x) = f (x)  \ny (x) = y h(x) + y p(x)  \ny h(x) = c 1y 1(x) + c 2y 2(x)  \nConstant Coe􀀎cients  \nay00 (x) + by0 (x) + cy(x) = f (x) y (x) = erx ) ar2 + br + c = 0  \nCases  \nDistinct, real roots: r = r 1 ;2; y h(x) = c 1er1 x + c 2er2 x  \nOne real root: y h(x) = (c 1 + c 2x)erx  \nComplex roots: r = 􀀋 􀀆 i􀀌; y h(x) = (c 1 cos 􀀌x + c 2 sin 􀀌x)e􀀋x  \nCauchy-Euler Equations  \nax2y00 (x) + bxy0 (x) + cy(x) = f (x) y (x) = xr ) ar (r 􀀀 1) + br + c = 0  \nCases  \nDistinct, real roots: r = r 1 ;2; y h(x) = c 1xr1 + c 2xr2  \nOne real root: y h(x) = (c 1 + c 2 ln j xj)xr  \nComplex roots: r = 􀀋 􀀆 i􀀌;  \ny h (x) = (c 1 cos(􀀌 ln j xj) + c 2 sin(􀀌 ln j xj))x􀀋  \nNonhomogeneous Problems  \nMethod of Undetermined Coe􀀎cients  \nf (x)  \nanxn + 􀀁 􀀁 􀀁 + a 1x + a 0 aebx  \nacos!x + bsin!x  \ny p(x)  \nAnxn + 􀀁 􀀁 􀀁 + A1x + A0 Aebx  \nAcos!x + B sin!x  \nModi􀀌ed Method of Undetermined Coe􀀎cients: if any term in the guess y p(x) is a solution of the homogeneous equation, then multiply the guess by xk , where k is the smallest positive integer such that no term in xky p(x) is a solution of the homogeneous problem.  \nReduction of Order  \nHomogeneous Case  \nGiven y 1(x) satis􀀌es L [y] = 0 ; 􀀌nd second linearly independent solution as v (x) = v(x)y 1(x): z = v0 satis􀀌es a separable ODE.  \nNonhomogeneous Case  \nGiven y 1(x) satis􀀌es L [y] = 0 ; 􀀌nd solution of L [y] = f as v (x) = v(x)y 1(x): z = v0 satis􀀌es a 􀀌rst order linear ODE.  \nMethod of Variation of Parameters  \ny p(x) = c 1(x)y 1(x) + c 2(x)y 2(x) c01(x)y 1(x) + c02(x)y 2(x) = 0  \nc01(x)y01(x) + c02(x)y02(x) = fa((xx))  \nApplications  \nFree Fall  \nx00 (t) = 􀀀g  \nv0 (t) = 􀀀g + f(v)  \nPopulation Dynamics  \nP0 (t) = kP (t)  \nP0 (t) = kP(t) 􀀀 bP2 (t)  \nNewton's Law of Cooling  \nT0 (t) = 􀀀k(T(t) 􀀀 Ta)  \nOscillations  \nmx00 (t) + kx(t) = 0  \nmx00 (t) + bx0 (t) + kx(t) = 0  \nmx00 (t) + bx0 (t) + kx(t) = F (t)  \nTypes of Damped Oscillation  \nOverdamped, b2 > 4mk Critically Damped, b2 = 4mk Underdamped, b2 \u003C 4mk  \nNumerical Methods  \nEuler's Method  \ny 0 = y (x 0) ;  \nyn = yn􀀀1 + 􀀁xf(xn􀀀1; yn􀀀1); n = 1; : : : ; N:  \nSeries Solutions Taylor Method  \nf (x) 􀀘 P cnxn ; cn = f~~ ~~(nn)!(0)  \n1. Di􀀋erentiate DE repeatedly.  \n2. Apply initial conditions.  \n3. Find Taylor coe􀀎cients.  \n4. Insert coe􀀎cients into series form for y (x):  \nPower Series Solution  \n1. Let y(x) =P cn(x 􀀀 a)n:  \n2. Find y0 (x); y00 (x):  \n3. Insert expansions in DE.  \n4. Collect like terms using reindexing.  \n5. Find recurrence relation.  \n6. Solve for coe􀀎cients and insert in y (x) series.  \nOrdinary and Singular Points  \ny00 + a(x)y0 + b(x)y = 0: x 0 is a  \nOrdinary point: a (x); b (x) real analytic in jx 􀀀 x 0j \u003C R Regular singular point: (x 􀀀 x 0)a (x), (x 􀀀 x 0)2 b (x) have convergent Taylor series about x = x 0:  \nIrregular singular point: Not ordinary or regular singular point.  \nFrobenius Method  \n1. Let y(x) =P cn(x 􀀀 x 0)n+r:  \n2. Obtain indicial equation r (r 􀀀 1) + a 0r + b 0:  \n3. Find recurrence relation based on types of roots of indicial equation.  \n4. Solve for coe􀀎cients and insert in y (x) series.  \nLaplace Transforms Transform Pairs  \nc  \nc  \neat  \ntn sin!t  \ncos!t  \nsinh at  \ncosh at  \nH (t 􀀀 a)  \n􀀎 (t 􀀀 a)  \ns 1  \ns 􀀀 a ,  \nn!  \nsn+1! ;  \n\n| s2 + !2\u003Cbr>s |\n| --- |\n| s2 + !2\u003Cbr>a","cbCaio4DJiBYfc1P","https://ap.wps.com/l/cbCaio4DJiBYfc1P","pdf",224476,"English","# First Order Equations\n## Separable\n## Linear First Order\n## Exact Equations\n## Non-Exact Form\n# Second Order Equations\n## Linear Second Order\n## Constant Coefficients\n## Cauchy-Euler Equations\n## Nonhomogeneous Problems\n### Undetermined Coefficients\n### Reduction of Order\n### Variation of Parameters\n# Applications and Numerical Methods\n## Free Fall\n## Population Dynamics\n## Newton's Law of Cooling\n## Euler's Method\n## Series Solutions\n# Ordinary and Singular Points\n## Frobenius Method\n# Laplace Transforms\n## Transform Pairs\n## Transform Properties\n## Solve Initial Value Problems\n# Special Functions and Systems\n## Legendre Polynomials\n## Bessel and Gamma Functions\n## Planar Systems and Eigenvalues","[{\"question\":\"How do you solve first-order linear differential equations using an integrating factor?\",\"answer\":\"Rewrite the equation as y'(x)+p(x)y(x)=f(x), compute the integrating factor μ(x)=exp(∫p(x)dx), then use (μy)'=μf to integrate and add the constant C.\"},{\"question\":\"What is the condition for an equation to be exact in the form M(x,y)dx+N(x,y)dy=0?\",\"answer\":\"Exactness requires My(x,y)=Nx(x,y). When this holds, introduce a potential u(x,y) so that du=Mdx+Ndy and set u(x,y)=const.\"},{\"question\":\"Which method is used for solving second-order nonhomogeneous ODEs and how is it selected?\",\"answer\":\"Use the method of undetermined coefficients when the forcing term matches standard guess patterns. For reduction of order, start from a known homogeneous solution y1. Use variation of parameters when writing the particular solution as y_p=c1(x)y1(x)+c2(x)y2(x) with auxiliary equations for c1' and c2'.\"}]","ODE Cheat Sheet - First Order Equations - Solution Methods | PDF",1791923998]