--,separableandhomogeneousequationsAsecond-orderlineardifferentialequationhastheform(1) whereP,Q,R,-OrderLinearEquationsIfG(x)=0forallx,suchequationsarecalledsecond-orderhomogeneouslinearequations.(.)(2)Ifforsomex,,binationisalsoasolution.(3)TheoremIfandarebothsolutionsofthelinearequation(2)andandareanyconstants,,,ifneitherxnoryisaconstantmultipleoftheother,,thefunctionandarelinearlydependent,.(4)TheoremIfandarelinearlyindependentsolutionsofEquation2,,itisnoteasytodiscoverparticularsolutionstoasecond-,QandRareconstantfunctions,thatis,ifthedifferentialequationhastheform(5),theexponentialfunctionybecauseitsderivativesareconstantsmultipleofitself:.SubstitutetheseexpressionintoEquation5Noticeisnever0soisasolutionofEquation5ifrisarootoftheequation(6)whichiscalledtheauxiliaryequation(orcharacteristicequation),therootandoftheauxiliaryequationcanbefound:(7),soandaretwolinearlyindependentsolutionsofEquation5.(8)Iftherootsandoftheauxiliaryequation`arerealandunequal,thentheg
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