ylto B=wdfqtky=fH yusssi.eattlso.se#ale7a.ti .GL#=vIT+wTascwt-dtce-ft ypfttceht e-ktftessfqdstce.tt daftly )'=ehtfh deftly etty = ftesflslds t C Then

Transcription

1 Math 307 Week 4

2 Mon Oct 15 Example Review : dfattsysttttbioardifpeg )'ehtfh Lefty integrating factor etty ftesflslds t C Then ektftessfqdstcett y # daftly yay cos C wt ) ypfttceht Try yp # Aaolwttt Bsincwt ) ' Lp H tkypttt wlasincwtttbcoscwtdttbca cos Cutt Bsinlwtl ) nitial value problem fwatkb ) sincwtlscodwt : BwdfqtkyfH As yaw ' ' either { ylto ) yo ht coscwt 91 GL#vT+wTascwtdtceft yusssieattlsose#ale7ati ypasse#wascwtltsywssinlwdyityektftehsflddstyoe 6 where tomy w Exercise Shawtht Consequently s He solution atte V P is { deftly eat y# a syo k4

3 Find Entangle the general lution of the diff eg def they E ht ho ( estyj ebtehtsl etty t t C y ts test + CEST $

4 YH y Special case : The general case : (1) of thy ht ftsds TH n draft plays pit cmamoas Let AH ftplslds Htt N so B 'HpH Lefty ehttkl Then a is est ya ' flsldstc ( et 't ' ya ) ' quintet et 't 't # to guys e ttfehsadstcett et " ya fte ' foods t C yet ) e Pitt femtcssdstceph Note this is of the form : ypltltcyhtt particular hn cliff folium eg of homogeneous 't pity o

5 Eixample hethe initial value problem y 't3tyte ' sod : nitial condition BHsf%ds3gt ' glass # te 4 test e 6 ' ice ' ytttfeticet ce45 fessttdt " sfe4tce3ki@sthgje 5e6 je so etty e%t2+c ft ) ( Gen solution ) d # Eft 2W e T

6 This is a linear diff eg : t µ 024 C5ottPQHs dude t ntegrating factor : B # 3h15Ott ) EPH! e3ht50*) ( 50*30 * 024 Q # to 650Mt c : Qld t fo cgott QH ) 6150ft ) t Thetas is full wear } Vfd 100+2* sds c Ott 245 csottldt at that time Choo )Q %}8% lbygae )3

7 lve Wed lt Oct Eixample the following nitial value problem : lution Write in teetotum The initial condition dfat? yt ycdl ftpsgsdsshlltt4 yields the equation Than BH eat ) elnllttylt#4 " hee ltt ( )yajchtdt 't ntegrate to get ( tt Csl ' Hence )yh fcittttat 4 or yal 4 a ylt HT 4 C 4 HUH CTY 4 Ht

8 Let lution be the Eixample following quiets Consider the L reeler circuit : A5 nvm or current measured clockwise Helly a Then law shows : voltage L t R 9 The Kirchoff voltage Vr across to resistor is R At time t so see the switch is a Consequently Vr lution d shown is closed auth the VP Voltage the increases across the resistor as shown on dkqtts9 Koto following graph : R On ( since Rs daff tf 9 Koto lving gives : THS 9 g e Ft What is the inductance L of the Set t Lf to get! 94 et) inductor in the circuit? * Vratsa 569

9 E Tn Tatts 9C e t4so) RTE ' ) A 450 fo L X lot H Henries ' 10µA

10 Examples solution Consider the electrical TEH is the solution circuit of the R 100 r initial value problem lov c? RC dug t k { at The capacitor is unchanged Taco ) 0 At time t O seconds the switch is closed and the The liton is voltage TEH across the y # to ( e HRC the increases ) capiacitvr # we as shown in the following to ( e graph : TG ood loll " ) e T 63 From the graph 8 4 T 2 8 t TE 3 What is the value of C? 42 t 00 C O See C O Too 10µF Farads

11 lution let w 22 TH s toes wine as hot y) O e bit w est where ton Y wth For t large we ignore the exponential : can TH lots 50 Cut We has to choose is so that f g S2 y) diff w satire hi '

12 htt VH That Remark Note the similarity : duets TA Newton 's Law of Cooling Tr d s LE uan R a dtfrsrtk ) vain Tr R * du Freefall with air resistance Bmw at Not an* m Nf 2mg aya k ( terminal velocity)

13 Let ( KS rl effort Fri Oct 19 reinter Changing the time measured in Tonichdli wmg minutes D whine H Let s he time measured in seconds Suppose yisa quantity that changes the differential img imauodiyt da!t2ddft3y costs) radius HN d dfs Find a diff htt k > o Express this interns oft rattertlans From the homework) a spherical RC circuits raindrop evaporates according tottefdloiydjpqm + d s > o where vvdy chip djgeqmtnvisardrintemsy Sssuyaaaoa r radius Find asinge dependent variable Find Htt c R sak my RtqgNT dip egnsfntfandtc

14 BigPe!Tme \ Autonomous hdntihtfltmynh soames Flyeto fhgl fixed Emuleisodctodtntitnthdf7T?g ( point fthkldtsftglsldstcdftpthysn stage dyagt hlyl hn big h : dit Ce ytdyphtt Blasted"4dds Lp H s e H ftpcsld dfftkyshjlhes/tehsfcsldsdff+sga f t cost ykkagtcelstdfatbyascat/jhacoscwtyltceht

15 O r Rcimt R to C i ooo Lct Live date tir + : Vr Thus A) 33 ( 106 x Farads time constant RC to Tee to Eth ) micro Ucs 208ns 33T D 105 m see