Differential Equations - Basic Concepts

The most general linear second order differential equation is in the form. ... to solve second order non-constant coefficient differential ... GoTo Notes PracticeandAssignmentproblemsarenotyetwritten.AstimepermitsIamworkingonthem,howeverIdon'thavetheamountoffreetimethatIusedtosoitwilltakeawhilebeforeanythingshowsuphere. Show/Hide ShowallSolutions/Steps/etc. HideallSolutions/Steps/etc. Sections SecondOrderDE'sIntroduction Real&DistinctRoots Chapters FirstOrderDE's LaplaceTransforms Classes Algebra CalculusI CalculusII CalculusIII DifferentialEquations Extras Algebra&TrigReview CommonMathErrors ComplexNumberPrimer HowToStudyMath CheatSheets&Tables Misc ContactMe MathJaxHelpandConfiguration NotesDownloads CompleteBook PracticeProblemsDownloads Problemsnotyetwritten. AssignmentProblemsDownloads Problemsnotyetwritten. 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Section3-1:BasicConcepts Inthischapterwewillbelookingexclusivelyatlinearsecondorderdifferentialequations.Themostgenerallinearsecondorderdifferentialequationisintheform. \[\begin{equation}p\left(t\right)y''+q\left(t\right)y'+r\left(t\right)y=g\left(t\right)\label{eq:eq1}\end{equation}\] Infact,wewillrarelylookatnon-constantcoefficientlinearsecondorderdifferentialequations.Inthecasewhereweassumeconstantcoefficientswewillusethefollowingdifferentialequation. \[\begin{equation}ay''+by'+cy=g\left(t\right)\label{eq:eq2}\end{equation}\] Wherepossiblewewilluse\(\eqref{eq:eq1}\)justtomakethepointthatcertainfacts,theorems,properties,and/ortechniquescanbeusedwiththenon-constantform.However,mostofthetimewewillbeusing\(\eqref{eq:eq2}\)asitcanbefairlydifficulttosolvesecondordernon-constantcoefficientdifferentialequations. Initiallywewillmakeourlifeeasierbylookingatdifferentialequationswith\(g(t)=0\).When\(g(t)=0\)wecallthedifferentialequationhomogeneousandwhen\(g\left(t\right)\ne0\)wecallthedifferentialequationnonhomogeneous. So,let’sstartthinkingabouthowtogoaboutsolvingaconstantcoefficient,homogeneous,linear,secondorderdifferentialequation.Hereisthegeneralconstantcoefficient,homogeneous,linear,secondorderdifferentialequation. \[ay''+by'+cy=0\] It’sprobablybesttostartoffwithanexample.Thisexamplewillleadustoaveryimportantfactthatwewilluseineveryproblemfromthispointon.Theexamplewillalsogiveuscluesintohowtogoaboutsolvingtheseingeneral. Example1Determinesomesolutionsto \[y''-9y=0\] ShowSolution Wecangetsomesolutionsheresimplybyinspection.Weneedfunctionswhosesecondderivativeis9timestheoriginalfunction.OneofthefirstfunctionsthatIcanthinkofthatcomesbacktoitselfaftertwoderivativesisanexponentialfunctionandwithproperexponentsthe9willgettakencareofaswell. So,itlookslikethefollowingtwofunctionsaresolutions. \[y\left(t\right)={{\bf{e}}^{3t}}\hspace{0.25in}\,\,\,\,\,\,{\mbox{and}}\hspace{0.25in}y\left(t\right)={{\bf{e}}^{-3t}}\] We’llleaveittoyoutoverifythattheseareinfactsolutions. Thesetwofunctionsarenottheonlysolutionstothedifferentialequationhowever.Anyofthefollowingarealsosolutionstothedifferentialequation. \[\begin{align*}y\left(t\right)&=-9{{\bf{e}}^{3t}}\hspace{0.25in}\hspace{0.25in}y\left(t\right)=123{{\bf{e}}^{3t}}\\y\left(t\right)&=56{{\bf{e}}^{-3t}}\hspace{0.25in}\hspace{0.25in}y\left(t\right)=\frac{{14}}{9}{{\bf{e}}^{-3t}}\\y\left(t\right)&=7{{\bf{e}}^{3t}}-6{{\bf{e}}^{-3t}}\hspace{0.25in}y\left(t\right)=-92{{\bf{e}}^{3t}}-16{{\bf{e}}^{-3t}}\end{align*}\] Infactifyouthinkaboutitanyfunctionthatisintheform \[y\left(t\right)={c_1}{{\bf{e}}^{3t}}+{c_2}{{\bf{e}}^{-3t}}\] willbeasolutiontothedifferentialequation. Thisexampleleadsustoaveryimportantfactthatwewilluseinpracticallyeveryprobleminthischapter. PrincipleofSuperposition If\({y_1}\left(t\right)\)and\({y_2}\left(t\right)\)aretwosolutionstoalinear,homogeneousdifferentialequationthensois \[\begin{equation}y\left(t\right)={c_1}{y_1}\left(t\right)+{c_2}{y_2}\left(t\right)\label{eq:eq3}\end{equation}\] Notethatwedidn’tincludetherestrictionofconstantcoefficientorsecondorderinthis.Thiswillworkforanylinearhomogeneousdifferentialequation. Ifwefurtherassumesecondorderandoneothercondition(whichwe’llgiveinasecond)wecangoastepfurther. If\({y_1}\left(t\right)\)and\({y_2}\left(t\right)\)aretwosolutionstoalinear,secondorderhomogeneousdifferentialequationandtheyare“niceenough”thenthegeneralsolutiontothelinear,secondorderhomogeneousdifferentialequationisgivenby\(\eqref{eq:eq3}\). So,justwhatdowemeanby“niceenough”?We’llholdoffonthatuntilalatersection.Atthispointyou’llhopefullybelieveitwhenwesaythatspecificfunctionsare“niceenough”. So,ifwenowmaketheassumptionthatwearedealingwithalinear,secondorderhomogeneousdifferentialequation,wenowknowthat\(\eqref{eq:eq3}\)willbeitsgeneralsolution.Thenextquestionthatwecanaskishowtofindtheconstants\(c_{1}\)and\(c_{2}\).Sincewehavetwoconstantsitmakessense,hopefully,thatwewillneedtwoequations,orconditions,tofindthem. Onewaytodothisistospecifythevalueofthesolutionattwodistinctpoints,or, \[y\left({{t_0}}\right)={y_0}\hspace{0.25in}y\left({{t_1}}\right)={y_1}\] Thesearetypicallycalledboundaryvaluesandarenotreallythefocusofthiscoursesowewon’tbeworkingwiththemhere.Wedogiveabriefintroductiontoboundaryvaluesinalaterchapterifyouareinterestedinseeinghowtheyworkandsomeoftheissuesthatarisewhenworkingwithboundaryvalues. Anotherwaytofindtheconstantswouldbetospecifythevalueofthesolutionanditsderivativeataparticularpoint.Or, \[y\left({{t_0}}\right)={y_0}\hspace{0.25in}y'\left({{t_0}}\right)={y'_0}\] Thesearethetwoconditionsthatwe’llbeusinghere.Aswiththefirstorderdifferentialequationsthesewillbecalledinitialconditions. Example2SolvethefollowingIVP. \[y''-9y=0\hspace{0.25in}y\left(0\right)=2\,\,\,\,\,\,\,y'\left(0\right)=-1\] ShowSolution First,thetwofunctions \[y\left(t\right)={{\bf{e}}^{3t}}\hspace{0.25in}\,\,\,\,\,\,{\mbox{and}}\hspace{0.25in}y\left(t\right)={{\bf{e}}^{-3t}}\] are“niceenough”forustoformthegeneralsolutiontothedifferentialequation.Atthispoint,pleasejustbelievethis.Youwillbeabletoverifythisforyourselfinacoupleofsections. Thegeneralsolutiontoourdifferentialequationisthen \[y\left(t\right)={c_1}{{\bf{e}}^{-3t}}+{c_2}{{\bf{e}}^{3t}}\] Nowallweneedtodoisapplytheinitialconditions.Thismeansthatweneedthederivativeofthesolution. \[y'\left(t\right)=-3{c_1}{{\bf{e}}^{-3t}}+3{c_2}{{\bf{e}}^{3t}}\] Plugintheinitialconditions \[\begin{align*}2&=y\left(0\right)={c_1}+{c_2}\\-1&=y'\left(0\right)=-3{c_1}+3{c_2}\end{align*}\] Thisgivesusasystemoftwoequationsandtwounknownsthatcanbesolved.Doingthisyields \[{c_1}=\frac{7}{6}\hspace{0.25in}{c_2}=\frac{5}{6}\] ThesolutiontotheIVPisthen, \[y\left(t\right)=\frac{7}{6}{{\bf{e}}^{-3t}}+\frac{5}{6}{{\bf{e}}^{3t}}\] Uptothispointwe’veonlylookedatasingledifferentialequationandwegotitssolutionbyinspection.Forararefewdifferentialequationswecandothis.However,forthevastmajorityofthesecondorderdifferentialequationsouttherewewillbeunabletodothis. So,wewouldlikeamethodforarrivingatthetwosolutionswewillneedinordertoformageneralsolutionthatwillworkforanylinear,constantcoefficient,secondorderhomogeneousdifferentialequation.Thisiseasierthanitmightinitiallylook. Wewillusethesolutionswefoundinthefirstexampleasaguide.Allofthesolutionsinthisexamplewereintheform \[y\left(t\right)={{\bf{e}}^{r\,t}}\] Note,thatwedidn’tincludeaconstantinfrontofitsincewecanliterallyincludeanyconstantthatwewantandstillgetasolution.Theimportantideahereistogettheexponentialfunction.Oncewehavethatwecanaddonconstantstoourheartscontent. So,let’sassumethatallsolutionsto \[\begin{equation}ay''+by'+cy=0\label{eq:eq4}\end{equation}\] willbeoftheform \[\begin{equation}y\left(t\right)={{\bf{e}}^{r\,t}}\label{eq:eq5}\end{equation}\] Toseeifwearecorrectallweneedtodoisplugthisintothedifferentialequationandseewhathappens.So,let’sgetsomederivativesandthenplugin. \[y'\left(t\right)=r{{\bf{e}}^{r\,t}}\hspace{0.25in}\hspace{0.25in}y''\left(t\right)={r^2}{{\bf{e}}^{r\,t}}\] \[\begin{align*}a\left({{r^2}{{\bf{e}}^{r\,t}}}\right)+b\left({r{{\bf{e}}^{r\,t}}}\right)+c\left({{{\bf{e}}^{r\,t}}}\right)&=0\\{{\bf{e}}^{r\,t}}\left({a{r^2}+br+c}\right)&=0\end{align*}\] So,if\(\eqref{eq:eq5}\)istobeasolutionto\(\eqref{eq:eq4}\)thenthefollowingmustbetrue \[{{\bf{e}}^{r\,t}}\left({a{r^2}+br+c}\right)=0\] Thiscanbereducedfurtherbynotingthatexponentialsareneverzero.Therefore,\(\eqref{eq:eq5}\)willbeasolutionto\(\eqref{eq:eq4}\)provided\(r\)isasolutionto \[\begin{equation}a{r^2}+br+c=0\label{eq:eq6}\end{equation}\] Thisequationistypicallycalledthecharacteristicequationfor\(\eqref{eq:eq4}\). Okay,sohowdoweusethistofindsolutionstoalinear,constantcoefficient,secondorderhomogeneousdifferentialequation?Firstwritedownthecharacteristicequation,\(\eqref{eq:eq6}\),forthedifferentialequation,\(\eqref{eq:eq4}\).Thiswillbeaquadraticequationandsoweshouldexpecttworoots,\(r_{1}\)and\(r_{2}\).Oncewehavethesetworootswehavetwosolutionstothedifferentialequation. \[\begin{equation}{y_1}\left(t\right)={{\bf{e}}^{{r_{\,1}}\,t}}\hspace{0.25in}\,\,\,\,\,\,\,\,{\mbox{and}}\hspace{0.25in}{y_2}\left(t\right)={{\bf{e}}^{{r_{\,2}}\,t}}\label{eq:eq7}\end{equation}\] Let’stakealookataquickexample. Example3Findtwosolutionsto \[y''-9y=0\] ShowSolution Thisisthesamedifferentialequationthatwelookedatinthefirstexample.Thistimehowever,let’snotjustguess.Let’sgothroughtheprocessasoutlinedabovetoseethefunctionsthatweguessabovearethesameasthefunctionstheprocessgivesus. Firstwritedownthecharacteristicequationforthisdifferentialequationandsolveit. \[{r^2}-9=0\hspace{0.25in}\Rightarrow\hspace{0.25in}r=\pm\,3\] Thetworootsare3and-3.Therefore,twosolutionsare \[{y_1}\left(t\right)={{\bf{e}}^{3t}}\hspace{0.25in}\,\,\,\,\,\,{\mbox{and}}\hspace{0.25in}{y_2}\left(t\right)={{\bf{e}}^{-3t}}\] Thesematchupwiththefirstguessesthatwemadeinthefirstexample. You’llnoticethatweneglectedtomentionwhetherornotthetwosolutionslistedin\(\eqref{eq:eq7}\)areinfact“niceenough”toformthegeneralsolutionto\(\eqref{eq:eq4}\).Thiswasintentional.Wehavethreecasesthatweneedtolookatandthiswillbeaddresseddifferentlyineachofthesecases. So,whatarethecases?Aswepreviouslynotedthecharacteristicequationisquadraticandsowillhavetworoots,\(r_{1}\)and\(r_{2}\).Therootswillhavethreepossibleforms.Theseare Real,distinctroots,\({r_1}\ne{r_2}\). Complexroot,\({r_{1,2}}=\lambda\pm\mu\,i\). Doubleroots,\({r_1}={r_2}=r\). Thenextthreesectionswilllookateachoftheseinsomemoredepth,includinggivingformsforthesolutionthatwillbe“niceenough”togetageneralsolution.