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Mehdi Rahmani-Andebili Diff erential Equations Practice Problems, Methods, and Solutions Differential Equations Mehdi Rahmani-Andebili Differential Equations Practice Problems, Methods, and Solutions MehdiRahmani-Andebili ElectricalEngineeringDepartment MontanaTechnologicalUniversity Butte,MT,USA ISBN978-3-031-07983-2 ISBN978-3-031-07984-9 (eBook) https://doi.org/10.1007/978-3-031-07984-9 #TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whetherthewholeor part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageand retrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafter developed. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsand regulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbookarebelievedto betrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsortheeditorsgiveawarranty, expressedorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissionsthatmayhavebeen made.Thepublisherremainsneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Differential equations course is one of the main courses of all engineering majors which is taught for freshman orsophomorestudents. Thesubjects includedifferent types offirst-order differential equations, including separable differential equation, linear differential equation, Bernoulli differential equation, and complete (exact) differential equation; different types of second-order differential equations with constant coefficients, including homogeneous and non-homogeneousdifferentialequationsthataresolvedusinginverseoperatormethod,unde- termined coefficient method, and Lagrange method; and differential equations with variable coefficientsthataresolvedbyusingseries andLaplacetransform.Moreover,many examples about Laplace transform and inverse Laplace transform of different types of functions are presented. Like the previously published textbooks, the textbook includes very detailed and multiple methods of problem solutions. It can be used as a practicing textbook by students and as a supplementaryteachingsourcebyinstructors. To help students study the textbook in the most efficient way, the exercises have been categorizedinninedifferentlevels.Inthisregard,foreachproblemofthetextbookadifficulty level (easy, normal, or hard) and a calculation amount (small, normal, or large) have been assigned.Moreover,ineachchapter,problemshavebeenorderedfromtheeasiestproblemwith thesmallestcalculationstothemostdifficultproblemswiththelargestcalculations.Therefore, students are suggested to start studying the textbook from the easiest problems and continue practicing until they reach the normal and then the hardest ones. On the other hand, this classification can help instructors choose their desirable problems to conduct a quiz or a test. Moreover,theclassificationofcomputationamountcanhelpstudentsmanagetheirtimeduring futureexamsandinstructorsgivetheappropriateproblemsbasedontheexamduration. Sincetheproblemshaveverydetailedsolutionsandsomeofthemincludemultiplemethods ofsolution,thetextbookcanbeusefulfortheunder-preparedstudents.Inaddition,thetextbook isbeneficialforknowledgeablestudentsbecauseitincludesadvancedexercises. Inthepreparationofproblemsolutions,ithasbeentriedtousetypicalmethodstopresentthe textbookasaninstructor-recommendedone.Inotherwords,theheuristicmethodsofproblem solutionhaveneverbeenusedasthefirstmethodofproblemsolution.Byconsideringthiskey point,thetextbookwillbeinthedirectionofinstructors’lectures,andtheinstructorswillnot seeanyuntaughtproblemsolutionsintheirstudents’answersheets. The Iranian University Entrance Exams for the master’s and PhD programs in electrical engineering major is the main reference of the textbook; however, all the problem solutions havebeenprovidedbyme.TheIranianUniversityEntranceExamisoneofthemostcompeti- tiveuniversityentranceexamsintheworldthatallowsonly10%oftheapplicantstogetinto prestigiousandtuition-freeIranianuniversities. Butte,MT,USA MehdiRahmani-Andebili v The Other Works Published by the Author TheauthorhasalreadypublishedthebooksandtextbooksbelowwithSpringerNature. Textbooks Feedback Control SystemsAnalysisandDesign- Practice Problems, Methods,and Solutions, SpringerNature,2022. PowerSystemAnalysis–PracticeProblems,Methods,andSolutions,SpringerNature,2022. Advanced Electrical Circuit Analysis – Practice Problems, Methods, and Solutions, Springer Nature,2022. AC Electrical Circuit Analysis – Practice Problems, Methods, and Solutions, Springer Nature,2021. Calculus–PracticeProblems,Methods,andSolutions,SpringerNature,2021. Precalculus–PracticeProblems,Methods,andSolutions,SpringerNature,2021. DC Electrical Circuit Analysis – Practice Problems, Methods, and Solutions, Springer Nature,2020. Books Applications of Artificial Intelligence in Planning and Operation of Smart Grid, Springer Nature,2022. Design,Control,andOperationofMicrogridsinSmartGrids,SpringerNature,2021. ApplicationsofFuzzyLogicinPlanningandOperationofSmartGrids,SpringerNature,2021. OperationofSmartHomes,SpringerNature,2021. Planning and Operation of Plug-in Electric Vehicles: Technical, Geographical, and Social Aspects,SpringerNature,2019. vii Contents 1 Problems:First-OrderDifferentialEquations. . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 SolutionsofProblems:First-OrderDifferentialEquations. . . . . . . . . . . . . . . . 9 3 Problems:Second-OrderDifferentialEquations. . . . . . . . . . . . . . . . . . . . . . . . 29 4 SolutionsofProblems:Second-OrderDifferentialEquations. . . . . . . . . . . . . . 35 5 Problems:SeriesandTheirApplicationsinSolvingDifferentialEquations. . . 49 6 SolutionsofProblems:SeriesandTheirApplicationsinSolving DifferentialEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7 Problems:LaplaceTransformandItsApplicationsinSolving DifferentialEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 8 SolutionsofProblems:LaplaceTransformandItsApplications inSolvingDifferentialEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 ix About the Author MehdiRahmani-AndebiliisanassistantprofessorintheElectricalEngineeringDepartmentat MontanaTechnologicalUniversity,MT,USA.Beforethat,hewasalsoanassistantprofessorin theEngineeringTechnologyDepartmentatStateUniversityofNewYork,BuffaloState,NY, USA,during2019–2021.HereceivedhisfirstMScandPhDdegreesinelectricalengineering (powersystem) from Tarbiat ModaresUniversity and Clemson University in 2011 and 2016, respectively, and his second MSc degree in physics and astronomy from the University of AlabamainHuntsvillein2019.Moreover,hewasapostdoctoralfellowatSharifUniversityof Technologyduring2016–2017.Asaprofessor,hehastaughtmanycoursessuchasEssentials ofElectricalEngineeringTechnology,ElectricalCircuitsAnalysisI,ElectricalCircuitsAnaly- sisII,ElectricalCircuitsandDevices,IndustrialElectronics,RenewableDistributedGeneration andStorage,FeedbackControls,DCandACElectricMachines,andPowerSystemAnalysis. Dr.Rahmani-Andebilihasmorethan200single-authorandfirst-authorpublicationsincluding journalpapers,conferencepapers,textbooks,books,andbookchapters.HeisanIEEESenior Member and the permanent reviewer of many credible journals. His research areas include smart grid, power system operation and planning, integration of renewables and energy storagesintopowersystem,energyschedulinganddemand-sidemanagement,plug-inelectric vehicles, distributed generation, and advanced optimization techniques in power system studies. xi 1 Problems: First-Order Differential Equations Abstract In this chapter, different types of first-order differential equations, including separable differential equation, linear differentialequation,Bernoullidifferentialequation,andcomplete(exact)differentialequation,arestudied.Inthischapter, the problems are categorized in different levels based on their difficulty levels (easy, normal, and hard) and calculation amounts (small, normal, and large). Additionally, the problems are ordered from the easiest problem with the smallest computationstothemostdifficultproblemswiththelargestcalculations. 1.1. Solvethefollowingdifferentialequation. xy0(cid:2)2y¼x2 Difficultylevel ○Easy ●Normal ○Hard Calculationamount ●Small ○Normal ○Large 1) y¼cx2lnx+x2 2) y¼x2ex+cx2 3) y¼x2lnx+cx2 4) y¼cx2ex+x2 1.2. Calculatethesolutionofthedifferentialequationbelow. xdyþydx¼ sinxdx Difficultylevel ○Easy ●Normal ○Hard Calculationamount ●Small ○Normal ○Large 1) y¼c(cid:2) cosx 2) y¼1ðc(cid:2) cosxÞ x 3) y¼x(c(cid:2) cosx) 4) Noneoftheabove 1.3. Whatisthegeneralsolutionofthedifferentialequationbelow? ð2xyþxÞy0 ¼y Difficultylevel ○Easy ●Normal ○Hard Calculationamount ●Small ○Normal ○Large #TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 1 M.Rahmani-Andebili,DifferentialEquations,https://doi.org/10.1007/978-3-031-07984-9_1 2 1 Problems:First-OrderDifferentialEquations 1) y¼ce(cid:2)x2 2) y2e2y¼cx 3) y¼cx12ey 4) y¼ce(cid:2)x2y2 1.4. Inachemicalreaction,thefollowingdifferentialequationwiththeprimaryconditionsisgiven.Determinethevalueofk. dy 1 ¼(cid:2)ky, yðt ¼0Þ¼y , yðt ¼20Þ¼ y dt 0 2 0 Difficultylevel ○Easy ●Normal ○Hard Calculationamount ●Small ○Normal ○Large 1) 0.35 2) 0.035 3) 0.025 4) 0.25 1.5. Whatisthesolutionofthedifferentialequationbelowwiththegivencondition? xy y0 ¼ , yðxÞ6¼0,8x yþ1 Difficultylevel ○Easy ●Normal ○Hard Calculationamount ●Small ○Normal ○Large 1) 2y¼cx2 2) 2y¼clny2 3) 2y+ ln(cy)2¼x2 4) y¼cex2 1.6. Calculatethevalueofy(x¼1)inthedifferentialequationbelowwiththegivenprimarycondition. ðxþyÞdxþdy¼0, yðx¼0Þ¼0 Difficultylevel ○Easy ●Normal ○Hard Calculationamount ●Small ○Normal ○Large 1) e(cid:2)1 2) e 3) (cid:2)e 4) (cid:2)e(cid:2)1 1.7. Solvethedifferentialequationbelow. dy x (cid:2)3y¼x4 dx Difficultylevel ○Easy ●Normal ○Hard Calculationamount ●Small ○Normal ○Large 1) y¼x3(x+c) 2) y¼x2(x+c) 3) y¼x(x+c) 4) y¼ (cid:2)x3(x+c)