Chapter 3 Linear Equations of Higher Order (Page # 144)

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1 Ma Modr Dirial Equaios Lcur wk 4 Jul 4-8 Dr Firozzama Darm o Mahmaics ad Saisics Arizoa Sa Uivrsi This wk s lcur will covr har ad har 4 Scios 4 har Liar Equaios o Highr Ordr Pag # 44 Scio Iroducio: Scod Ordr Liar Equaios Pag # 44 A scod ordr dirial quaio i h ukow ucio is o o h orm G This dirial quaio is said o b liar rovidd ha G is liar i h dd variabl ad is drivaivs Th DE cos a is a scod ordr liar DE O h ohr had cos a is o a scod ordr liar FE bcaus o h o liar rm Th gral scod ordr liar DE has h orm A B F a Homogous Scod Ordr Liar Equaios: osidr h gral scod ordr liar quaio A B F whr h coici ucios A B ad F ar coiuous o h o irval I ad i h ucio F o h righ had sid vaishs idicall o I h h quaio a is said o b homogous liar quaio; ohrwis i is o-homogous Th scod ordr liar quaio cos is o-homogous; is associad homogous quaio is No ha h ohomogous rm F rqul corrsods o som ral iluc o h ssm Eaml Vri ha h ucios ad ar soluios o h dirial quaio ad h id a soluio saisig h iiial codiios ad Soluio: You ca asil vri ha ad ar soluios o h dirial quaio Now b h surosiio ricil Pag# 4 Thorm w kow ha h gral soluio is also hav ow c c c Th ollowig rsuls ar h oud: c c c c c c c c W c

2 Diiio: Liar Iddc o wo ucios: Two ucios did o a o irval I is said o liarl idd o I rovidd ha ihr is a cosa mulil o h ohr Diiio: Giv wo ucios ad q h Wroskia W q o ad q is h drmia q W W q q q q Thorm: Wroskias o soluios: Suos ha ad ar wo soluios o h homogous scod ordr liar quaio q o a o irval I o which ad q ar coiuous a I ad ar liarl dd h h Wroskia W o I b I ad ar liarl idd h h Wroskia W a ach oi o I Hom work roblms: Pag # 8 Scio Gral Soluios o Liar Equaios Pag # 8 Diiio: Liar Ddc o Fucios: Th ucios o b liarl dd o I rovidd ha hr iss cosas zro such ha c c c c o I or all ar said c c c c o all Eaml Show ha h ucios si si cos ar liarl dd b usig Wroskia Eaml Show ha h ucios liarl idd Eaml Show ha 4 cos si ar cos si is a soluio o Hom work roblms: Pag # 8 9 Scio Homogous Equaios wih osa oicis Pag # r haracrisic quaio or idig gral soluio: Suos b a soluio o h homogous quaio a a a a a wih cosa coicis a a a a a h ar a r ar ar a is calld h characrisic quaio or auiliar quaio o h DE Th soluio o DE is rducs o a soluio o a url algbraic quaio

3 Eaml Solv h iiial valu roblm ; Soluio: L r b h soluio o h iiial valu roblm Th characrisic quaio o his DE is r r r r B h abov horm h gral soluio is c c c c c c Th aricular soluio is h Usig iiial codiio o ca id ha Hom work roblms: Pag # Scio Nohomogous Equaios ad Udrmid oici Pag # 9 Th gral ohomogous -h ordr liar quaio wih cosa coicis has h orm a a a a a b has h gral soluio o h orm c whr h comlmar ucio c is a gral soluio o h associad homogous quaio a a a a a ad is h aricular soluio o b For idig h aricular soluio o b w d o mak a illig guss L id aricular soluio: Eaml Fid h aricular soluio o 4 Our DE is ohomogous o h orm b whr is olomial o dgr so our guss is A B h A will sais h DE rovidd ha A 4 A B A / 4 B / W hav h aricular soluio / 4 / Scio Forcd Oscillaios ad Rsoac Pag # 9 I his scio w hav h scod ordr DE m c k F ha govrs h o dimsioal moio o mass m ha is aachd o a srig wih cosa k ad a dasho wih cosa c ad is also acd o b a ral orc F Machis wih roaig comos commol ivolv mass-srig ssm or hir quivals i which h ral orc is siml harmoic: F F cosω or F F siω whr h cosa F is h amliud o h riodic orc ad ω is is circular rquc Udamd orc oscillaios: To sud h udamd oscillaios udr h iluc o h ral orc F F cosω ad sig c w hav m k F cosω No ha i has comlmar ucio k cosω siω wh ω is h m c

4 circular aural rquc o h mass-srig ssm O ca id h aricular F soluio o h ssm qual o cosω m ω ω Eaml osidr h DE m k F Also giv ha m k 9 F 8 adω Fid h gral soluio i Scio 8 Edoi Problms ad Eigvalus Pag # 9 W ar amiliar wih h ac ha a soluio o a scod-ordr liar DE is uiqul drmid b wo iiial codiios I his scio w will s ha h siuaio is radicall dir or a roblm such as q ; a b Th giv codiios ar a h d ois o h irval a b Such a roblm is calld a doi or boudar valu roblm I aricular q ; a b has h rivial soluio Eaml osidr h boudar valu roblm ; π I is as o s ha h gral soluio is Acos B si Usig d ois w id A B Thus h ol rivial soluio is Eaml osidr h boudar valu roblm 4 ; π I is as o s ha h gral soluio is Acos B si Usig d ois w id A ad or all valus o B w hav h orivial soluio is B si har 4 Iroducio o Ssms o Dirial Equaios Pag # 4 Scio 4 Firs Ordr Ssms ad Alicaios Pag # 4 I his scio w will rsric our aio o ssms i which h umbr o quaios is h sam as h umbr o dd variabls Firs ordr ssms: I cas o a ssm o wo scod ordr quaios w cosidr h orm Th irs ordr ssm cosisig o h sigl -h ordr quaio whr w iroduc h ollowig dd variabls Eaml osidr h hird ordr ssm o liar DE si This roblm is o h orm si Th subsiuios will

5 b si o h hr irs ordr quaios Eaml osidr h scod ordr ssm o liar DE 4si Trasr his ssm io a quival irs ordr ssm W us o obai h ollowig 4si o irs ordr quaios i h dd variabls Scio 4 Th Mhod o Elimiaio Pag # 4 Th mhod o limiaio or liar dirial ssms o quaios is similar o h soluio o a liar ssm o algbraic quaios b a rocss o limiaig h ukows a a im ol a sigl quaio wih a sigl ukow rmais Eaml Fid h aricular soluio o h ssm 4 ha saisis h iiial codiios B h mhod o limiaio w id Thus w hav 4 4 Usig characrisic quaio o ca id h gral soluio o h homogous ssm as Now usig iiial codiio w hav h dsird soluio

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