Differential Equations: Practice Problems, Methods, and Solutions

109 Pages·2022·1.67 MB·English

by  Mehdi Rahmani-Andebili

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
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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)