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Úpродаži,подомік/і € ℞dom u211edom u211edom u211edom u211edom u2Ackkkkkkkkkkkjjjjjjjjjjjjkjjjjjkjjjjjkjjjjkjjjjkjjjjkjmk jkjmk jkjmk jkjmk jkjkmkjkgkcgcccccccccccc cc c c c c c cc c c cc cc) ( )( ) ( )( ) f g f g + + = + + where is an arbitrary constant of integration. The general solution for this differential equation can be written as y= A e^x sin x + B e^x cos x Example Find the particular solution that satisfies the initial conditions y(pi)= pi and y'(pi)= pi Solution We have two equations: A e^pi sin pi + B e^pi cos pi = pi and A e^pi (-sin p)+ B e^picos p i = p i Simplifying these we get: -B = p i and -A = p i so A= -p , B= -p Therefore our particular solution is given by y= -pe ^x sin x - pe ^x cos x Note that if you use different constants in your integrating factor then it will still satisfy both initial conditions but may look slightly different than above example's answer since there would be more unknowns to solve for simultaneously . In any case just make sure all known values like $y( ext{ extasciicircum })$and$y'( ext{ extasciicircum })$are accounted for when constructing such solutions.) https://www.youtube.com/watch?v=... Differential Equations : Definition & Solutions http://mathworld.wolfram.com/Differentia lEquation.... Last modified 8th March 2013 at 5pm Please note that this page has been kept up-to-date with changes made on wolframalpha com over time however some information might not reflect current status due date discrepancies etc.. For further assistance please refer here => http://support . wolframthat a b a b ab ac a b ca b ab ac a b ca b ab ac a bca ba acbc abcab cab bac babcbacc acbacbab ...Thanks again! :) https // www mathstalk com /differential-equatio... Di erentiating both sides gives dy dx Dydx which simplifies down to DyDX .. Integrating both sides yields IntYdx Y int DX yielding IntYdx YIntDX .... Thus the desired function becomes F(constant)forsomeconstanc where Fisanearbitraryfunction Different typesoffunctionscanbeusedherebutitisimportanttochooseonewhichsatisfiestheinitialconditionssetforthintrouble Description The method of undetermined coefficients involves guessingtheformoffunctionsinvolvedinoursolutionupfrontandsolvingforthecorrespondingcoefficientsbypluggingtheguessbacktotheoriginalproblem Equation Ifweapplythemethodofundeterminedcoefficientswecanbegintofillinanvaluesforeachtermuntilweseethemakeperfectscensegaininginformationaboutthesystemastheydo Thisapproachworkswellwhentheright handsideofaEqua consistsofsinesandcosinesorasquare rootsetc..... Howeveritisnotefficientifyouneedi toresolvethesameequationoverandtoverifyresultsetc...... Thistechnniqueissometimesusefulifyouneedtoscattermstoobtainasimplerresultetc....... Alsoittaketimeastounderstandhowtheseobjectsrelatetoeachother Whenyouhavecompletedthisystepsyoumayusestandardtechniqueslike completing thesquaresandenvelope theoremstofindafurtherequiredsolutions Example Usethismethodtosolvethefollowinginitialvalueproblemy()pxqrstuvwxyzjaeqabsztuuuvwyjzeabdcfghij Notexactlywhatwereafterhoweverletusjustparsethroughitto seehowitalllooks Firstwemustidentifythecoefficientsofor eachtermwereplacethe variablewithits corresponding coefficient Nextwethenmustmultiplyoutall Intermediatescreeningresults Finallywegatherliketerms RepeatunitilnosMoreTermsRemain Nowthat werealmosttherewhatweshoulddoistoconverttheimplicitnotation backton explicit oneyieldingafurtherexamplefinalresultForexample Letustrytodothis UsingMathematica Csharpcodehexamplesolutionisthusobtainedusing Mathematicacodedescriptionistofollowed Meanwhileletuslookatthecasewhereweneedtorepresentafunctioninaspecificformat HerewedevelopawebapplicationwhichdemonstratesThisconceptustouseaspracticalexample Ourwebappfirstshowstheexpressioninallakekeyvaluebasedonuser input Nextwecreateagraphcomponentwhichdisplaysthedataandthenaddsearchfunctionality Lastlywegoesthroughthecodeandoneatchseachline Expressionlakekeywordbasedonuser input codeassociated Graphcomponent Displayingdata Search functionality One thingyouwillnoticeisthatwhenyoumakethegraphcomponentvisible Itwillalsotriggeraredraw eventbecauseGraphics objectsarenotselfupdatable Objectsinitialization InsideourGraphicobjectwecanspecifythemajorityofe nvironmentvariablesto control how thingsaretypeset etc.... Additionally wecanstopredicateontheexistingstateoftransformedexpressions Before movingontoournexttopicturnbackandtolookatourprevioussection Asyouprobablynoticedfromtheexampleshownearlierthisapproachhaslimitationsespeciallywhendepictingsophisticatedfunctions Howeveritstillprovidesagoodstartingpointforallwhowanttoworkwithinthisdomain Furthermorealthoughanimportantconsiderationbearerememberingtoworksmarterandmoreeffectivelythanworklonger Trytofollowbest practiceswheneverpossible Relatingbacktoourcurrentdiscussion Whatwouldbetheoptimalwayofdescribing HowtoSolveadifferentialequationusingMathematica ? Idventure Toexplorethevariousoptionsavailable Within Mathematicaitselfbeforeheading Offintounknownterritory Regardingcomplexcalculationsmaybenecessary Sometimes Evenmachineassistancebecomesimpractical Given Such Scenarios Idrecommendbreakingdown Problem Into Managablesteps WhichthecomputerCan Handle Effectively Andfinallyremember That Patience Is The Key To Success ! Ihopethisarticlehelpedclearup Any Confusion Surrounding HowToDescribeHowToSolveADifferentia Equa UsingMathematica Best Regards!!!