y'-2y=3x-2 How about yh and yp. (yh=?,yp=?) | Socratic

y(x)=Ae−x+Be2x−32x+74. Explanation: We have: y''−y'−2y=3x−2. This is a second order linear non-Homogeneous Differentiation Equation. #Y"-y'-2y=3x-2#Howaboutyhandyp.(yh=?,yp=?) Calculus 1Answer SteveM Oct10,2017 #y(x)=Ae^(-x)+Be^(2x)-3/2x+7/4# Explanation: Wehave: #y''-y'-2y=3x-2# Thisisasecondorderlinearnon-HomogeneousDifferentiationEquation.Thestandardapproachistofindasolution,#y_c#ofthehomogeneousequationbylookingattheAuxiliaryEquation,whichisthequadraticequationwiththecoefficientsofthederivatives,andthenfindinganindependentparticularsolution,#y_p#ofthenon-homogeneousequation. ComplementaryFunction Thehomogeneousequationassociatedwith[A]is #y''-y'-2y=0# Andit'sassociatedAuxiliaryequationis: #m^2-m-2=0# #(m-2)(m+1)=0# Whichhastworealanddistinctsolutions#m=-1,2# Thusthesolutionofthehomogeneousequationis: #y_c=Ae^((-1)x)+Be^((2)x)# #\\\=Ae^(-x)+Be^(2x)# ParticularSolution Withthisparticularequation[A],aprobablesolutionisoftheform: #y=ax+b# Where#a,b,c#areconstantstobedeterminedbysubstitution Letusassumetheabovesolutionworks,inwhichcasebedifferentiatingwrt#x#wehave: #y'\\=a# #y''=0# SubstitutingintotheinitialDifferentialEquation#[A]#weget: #0-a-2(ax+b)=3x-2# #:.-a-2ax-2b=3x-2# Equatingcoefficientsof#x^0#and#x#weget: #x^0:-a-2b=-2# #x^1:-2a=3# Solvingsimultaneouswehave: #a=-3/2,b=7/4# AndsoweformtheParticularsolution: #y_p=-3/2x+7/4# GeneralSolution WhichthenleadstotheGSof[A} #y(x)=y_c+y_p# #\\\\\\\=Ae^(-x)+Be^(2x)-3/2x+7/4# Aswehavealinearcombinationofthreelinearlyindependentsolutions,thisistheGS. Answerlink Relatedquestions HowdoIdeterminethemolecularshapeofamolecule? Whatisthelewisstructureforco2? Whatisthelewisstructureforhcn? Howisvseprusedtoclassifymolecules? Whataretheunitsusedfortheidealgaslaw? HowdoesCharle'slawrelatetobreathing? Whatistheidealgaslawconstant? Howdoyoucalculatetheidealgaslawconstant? Howdoyoufinddensityintheidealgaslaw? Doesidealgaslawapplytoliquids? Impactofthisquestion 4177views aroundtheworld Youcanreusethisanswer CreativeCommonsLicense