Solution Of A Differential Equation -General and Particular

When the arbitrary constant of the general solution takes some unique value, then the solution becomes the particular solution of the equation. CheckoutJEEMAINS2022QuestionPaperAnalysis: CheckoutJEEMAINS2022QuestionPaperAnalysis: × DownloadNow MathsMathArticleSolutionOfADifferentialEquation SolutionOfADifferentialEquation SolvingDifferentialEquations Thesolutionofadifferentialequation–Generalandparticularwilluseintegrationinsomestepstosolveit.Wewillbelearninghowtosolveadifferentialequationwiththehelpofsolvedexamples.Alsolearntothegeneralsolutionforfirst-orderandsecond-orderdifferentialequation.Letusfirstunderstandtosolveasimplecasehere: Considerthefollowingequation: 2x2–5x–7=0. Thesolutiontothisequationisanumberi.e.-1or7/2 whichsatisfiestheaboveequation.Whichmeansputtingthevalueofvariablexas-1or7/2,wegetLeft-handside(LHS)equaltoRight-handside(RHS)i.e0.Butinthecaseofthedifferentialequation,thesolutionisafunctionthatsatisfiesthegivendifferentialequation.Thatmeansweneedtodifferentiatethegivenequationfirstandthenfindthesolutionsforit.Thedifferentialequationsexaminedareoftheformy’=/(x,y)(equationsofhigherorderscouldbereducedtoequationsofthefirstorder).Thefunctionfisconsideredtobeanalyticinaadequatelylargeneighbourhoodoftheinitialpoint(x0,y0). Also,read:  LinearDifferentialEquation  FormationofDifferentialEquations  FirstOrderDifferentialEquation  DifferentialEquation–OrderandDegree DifferentialEquationsForClass12 DifferentialEquationsWorksheets DifferentialEquationsPdf Second-OrderDerivative OrdinaryDifferentialEquations PartialDifferentialEquation GeneralSolutionofaDifferentialEquation Whenthearbitraryconstantofthegeneralsolutiontakessomeuniquevalue,thenthesolutionbecomestheparticularsolutionoftheequation. Byusingtheboundaryconditions(alsoknownastheinitialconditions)theparticularsolutionofadifferentialequationisobtained. So,toobtainaparticularsolution,firstofall,ageneralsolutionisfoundoutandthen,byusingthegivenconditionstheparticularsolutionisgenerated. Suppose, dy/dx=ex+cos2x+2x3 Thenweknow,thegeneralsolutionis:  y=ex+sin2x/2+x4/2+C Now, x=0,y=5 substitutingthisvalueinthegeneralsolutionweget, 5=e0+sin(0)/2+(0)4/2+C  C=4  Hence,substitutingthevalueofCinthegeneralsolutionweobtain, y=ex+sin2x/2+x4/2+4 Thisrepresentstheparticularsolutionofthegivenequation. GeneralSolutionforFirstOrderandSecondOrder Ifwehavetosolveafirst-orderdifferentialequationbyvariableseparablemethod,weneedhavetomentionanarbitraryconstantbeforewestartperformingintegration.Hence,wecanseethatasolutionofthefirst-order differentialequationhasatleastonefixedarbitraryconstantaftersimplification. VariableseparabledifferentialEquations: Thedifferentialequationswhicharerepresentedintermsof(x,y)suchasthex-termsandy-termscanbeorderedtodifferentsidesoftheequation(includingdeltaterms).Thus,eachvariableafterseparationcanbeintegratedeasilytofindthesolutionofthedifferentialequation.Theequationscanbewrittenas: f(x)dx+g(y)dy=0,wheref(x)andg(y)areeitherconstantsorfunctionsofxandyrespectively. Similarly,thegeneralsolutionofasecond-orderdifferentialequationwillconsistoftwofixedarbitraryconstantsandsoon.Thegeneralsolutiongeometricallyinterpretsanm-parametergroupofcurves. ParticularSolutionofaDifferentialEquation AParticularSolutionisasolution ofadifferentialequationtakenfromtheGeneralSolutionbyallocatingspecificvaluestotherandomconstants.TherequirementsfordeterminingthevaluesoftherandomconstantscanbepresentedtousintheformofanInitial-ValueProblem,orBoundaryConditions,dependingonthequery. SingularSolution TheSingularSolutionof agivendifferentialequationisalsoatypeofParticularSolutionbutitcan’tbetakenfromtheGeneralSolutionbydesignatingthevaluesoftherandomconstants. DifferentialEquationsExample Example:dy/dx=x2  Solution:dy=x2dx Integratingbothsides,weget \(\begin{array}{l}\Rightarrow\intdy=\intx^2dx\end{array}\) Ifwesolvethisequationtofigureoutthevalueofyweget  y=x3/3+C  whereCisanarbitraryconstant. Intheabove-obtainedsolution,weseethaty isafunctionofx.Onsubstitutingthisvalueofyinthegivendifferentialequation,boththesidesofthedifferentialequationbecomesequal. DifferentialEquationsPracticeProblemswithSolutions Thesolutionobtainedaboveafterintegrationconsistsofafunctionandanarbitraryconstant.Thisrepresentsageneralsolutionofthegivenequation. Letthesolutionberepresentedas\(\begin{array}{l}y=\phi(x)+C\end{array}\).Itrepresentsthesolutioncurveortheintegralcurveofthegivendifferentialequation. Thus,wecansaythatageneralsolutionalwaysinvolvesaconstantC. Letusconsidersomemoreexamples: SolvedExamples Togetabetterinsightintothetopic,letushavealookatthefollowingexample. Example–Findouttheparticularsolutionofthedifferentialequationlndy/dx=e4y+lnx,giventhatforx=0,y=0. Solution–dy/dx=e4y+lnx dy/dx=e4y×elnx dy/dx=e4y×x  1/e4ydy=xdx e-4ydy=xdx  Integratingboththesideswithrespecttoyandxrespectivelyweget, e−4y/−4=x2/2+C Thisrepresentsthegeneralsolutionofthedifferentialequationgiven. Now,itisalsogiventhat y(0)=0,substitutingthisvalueintheabovegeneralsolutionweget, e0/−4=02/2+C ⇒C=−1/4 Hence,theaboveequationcanberewrittenas e−4y/−4=x2/2–14 ⇒e−4y=−2x2+1 ⇒ln(e−4y)=ln(1−2x2) ⇒−4y=ln(1−2x2) ⇒y=–ln(1−2x2)/4 whichistheparticularsolutionofthedifferentialequationgiven. Tolearnmoreaboutthesolutionofadifferentialequation,downloadBYJU’S-TheLearningApp. TestyourknowledgeonSolutionOfADifferentialEquation Q5 PutyourunderstandingofthisconcepttotestbyansweringafewMCQs.Click‘StartQuiz’tobegin! Selectthecorrectanswerandclickonthe“Finish”buttonCheckyourscoreandanswersattheendofthequiz StartQuiz Congrats! VisitBYJU’SforallMathsrelatedqueriesandstudymaterials Yourresultisasbelow 0outof0arewrong 0outof0arecorrect 0outof0areUnattempted ViewQuizAnswersandAnalysis MATHSRelatedLinks DivisibilityRules SetOfRealNumbers SquareRootsList DifferenceBetweenCubeAndCuboid BasicTrigonometryFormulas SetOfPrimeNumbers ArithmeticProgressionExamples RomanNumbers1To50 DifferentAngles NumberSystem 2Comments Amrita July10,2019at11:26am WhatisthemeaningofDifferentialEquations? Reply mentor February14,2020at2:56pm InMathematics,adifferentialequationisanequationthatcontainsoneormorefunctionswithitsderivatives.Thederivativesofthefunctiondefinetherateofchangeofafunctionatapoint.Itismainlyusedinfieldssuchasphysics,engineering,biology,andsoon.Themainpurposeofdifferentialequationisthestudyofsolutionsthatsatisfytheequations,andthepropertiesofthesolutions.Oneoftheeasiestwaystosolvethedifferentialequationisbyusingexplicitformulas.Inthisarticle,letusdiscussthedefinition,types,methodstosolvethedifferentialequation,orderanddegreeofthedifferentialequation,ordinarydifferentialequationswithreal-wordexampleandthesolvedproblem. Reply LeaveaCommentCancelreply YourMobilenumberandEmailidwillnotbepublished.Requiredfieldsaremarked* * SendOTP DidnotreceiveOTP? 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