ENGI 3424 Appendix Formulæ Page A-01

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1 ENGI 344 Appdix Formulæ g A-0 ENGI 344 Egirig Mthmtics ossibilitis or your Formul Shts You my slct itms rom this documt or plcmt o your ormul shts. Howvr, dsigig your ow ormul sht c b vlubl rvisio xrcis i itsl. Thr is r too much mtril hr to it withi your lloctio. You r titld to o doubl-sidd sht o stdrd siz (8½" ") i tst d to two such shts i th il xmitio.. First Ordr ODEs M x, y N x, y 0 Sprbl i M x, y x gy d Nx, y uxvy Lir: h h x y Rx ; solutio y R C, whr h Rductio o ordr (missig y trm): d y To solv x Rx, d x d y Rplc by p d rplc Rductio o ordr (missig x trm): d y To solv Q y R, d x d y Rplc by p d rplc Numricl Mthods by by dp dp p Eulr s mthod to id RK4 lgorithm: y x giv st ordr ODE d yx : y y h x y k x, y,, 3 k x h, y h k 4 3 k x h y h k k x h y h k h y y k k k k o,

2 ENGI 344 Appdix Formulæ g A-0. Scod Ordr Lir ODEs [ d Q both costt]: Auxiliry qutio: Solv Q Complmtry uctio: 0 d y, Q y R x Rl distict roots (ovr-dmpd): x x yc A B Rl rptd roots (criticlly dmpd): yc A Bx x Complx cojugt pir o roots b j (udr-dmpd): x jbx jbx x yc A B C cos bx Dsi bx rticulr solutio by vritio o prmtrs: Lt yc A y B y th id W y y 0 y W yr, y y R y W u u, W y u y v y, W y R v W W v, th rticulr solutio by udtrmid coicits: kx kx I Rx, th try y c I I R x = ( polyomil o dgr ), th try y 0 y R, y = ( polyomil o dgr ), with ll ( + ) coicits to b dtrmid. R x = ( multipl o cos kx d/or si kx), th try y ccos kx d si kx But: i prt (or ll) o y is icludd i th C.F., th multiply y by x. Grl solutio: y y y, th C Iitil (or boudry) coditios complt solutio. or us Lplc trsorms (Chptr 3) or sris solutios (Chptr 5).

3 ENGI 344 Appdix Formulæ g A Som Ivrs Lplc Trsorms Fs () t Fs () t 0 s s ( ) ( ) 0 0 s F s s s F s s s s st s s s () t dt 0 t t ( )! t t t t! t s t H t si t cos t d dt d dt cos s s s s 3 s s s s s F( s) s 0 df ds t t si t si t t cos t 3 t si t tcos t t () d t t

4 ENGI 344 Appdix Formulæ g A-04 First shit thorm: with F s L t bt Th ivrs Lplc trsorm o F s b is t. Scod shit thorm: Th ivrs Lplc trsorm o s F s is t H t. Sitig proprty o th Dirc dlt uctio: d c ( ) c d ( t) ( t ) dt 0 c or d Covolutio: I th Lplc trsorms o uctios t d gt r F (s) d G (s) rspctivly, th th ivrs Lplc trsorm o H s F s Gs is th covolutio t ( ) ( ) ( ) ( ) h t g t g t d t g d g t 0 0 t Complx Numbrs: j z x j y r r cos j r si * j / j k/ z r x y, z x j y r, z r k y rg whr t, x * z zz z

5 ENGI 344 Appdix Formulæ g A rtil Dirtitio Chi rul: I y x x x d x g t t t,,, i,,, m th y y xi y x y x y x t x t x t x t x t d j i j j j j i y y y y i x x x x i i Uss iclud rts o chg; pproximt d rltiv chgs Grdit: 3 ˆ I, ˆ ˆ i j k d x y z x y z Rt o chg o i th dirctio o t poit is th dirctiol drivtiv D ˆ Jcobi (implicit mthod): Covrsio rom x, x, x3,, x to,, 3,, qutios x, x, x,, x 0. i 3 Fid ll dirtils Jcobi (xplicit mthod): Jcobi i u u u u did implicitly by d, th costruct th mtrix qutio A du du B. du Th Jcobi is x x x u u u x x x x, x,, x ABS dt u u u u, u,, u T dt B. dt A x x x u u u

6 ENGI 344 Appdix Formulæ g A-06 Mx-Mi: Chck ll poits: - o th domi boudry; - whr is udid; - whr is udid; - whr 0. Scod drivtiv tst (t poits whr 0 ): D xx yx xy yy D > 0 d xx 0 locl miimum D > 0 d xx 0 locl mximum D < 0 sddl poit D = 0 : tst ils. Lgrg Multiplirs: Idtiy uctio,, 3,, Idtiy costrit(s) x x x x to b mximizd or miimizd. g x, x, x3,, x k. Solv th systm o qutios g d g k [o costrit] or g h d g k, h c [two costrits] Solutio with smllst (lrgst) vlu o is th miimum (mximum). 5. Sris d th Squz thorm: I d i c b Evry boudd mootoic squc is covrgt. b covrg to th sm limit L c lso covrgs to th limit L. Th Rc to Iiity c! c 0 logb b or ll suicitly lrg. Divrgc Tst: lim 0 divrgs

7 ENGI 344 Appdix Formulæ g A-07 Gomtric sris: k r r S r S r k r r DNE r I xct sum is rquird, th try tlscopig sris or th gomtric sris. p-sris: p covrgs [bsolutly] i p > ; divrgs othrwis. Altrtig p-sris: p covrgs bsolutly or p covrgs coditiolly or 0 p divrgs or p 0 Compriso Tst: b is rrc sris whos covrgc or divrgc is kow. b covrgs b divrgs 0 b covrgs o coclusio b 0 o coclusio divrgs Limit Compriso Tst: lim 0 b b covrgs b divrgs covrgs o coclusio lim L 0 covrgs divrgs b lim b o coclusio divrgs Altrtig Sris Tst: I 0 d lim 0 th covrgs.

8 ENGI 344 Appdix Formulæ g A-08 Rtio Tst: most usul wh th grl trm icluds xpotil or ctoril ctor. I lim th is bsolutly covrgt. I lim th I lim th th tst ils is divrgt. Root Tst (sldom rquird; th rtio tst c b usd i its plc): I lim th I lim th I lim th th tst ils is bsolutly covrgt. is divrgt. owr Sris: bd is 0 Th rdius o covrgc o x c Usully b, d 0 R lim R lim b. Tylor Sris (Mcluri i c = 0): Biomil Sris: x 0 c x x c! k k Itrvl o covrgc: I or 0 or [ th sris trmits t k ] I, or 0 I, or 0 I, or d ot itgr, with bsolut covrgc t x. k! with coditiol covrgc t x. with divrgc t x. x k

9 ENGI 344 Appdix Formulæ g A-09 Som stdrd (quotbl) Mcluri sris: 3 4 x x x x x x x!! 3! 4! 0 k 0 k 4 k x x x cos x x k!! 4! k 0 k 3 5 k x x x si x x x k! 3! 5! 3 4 x x x x l 3 4 x x x Error stimtio Altrtig sris: Tylor sris: whr is som umbr btw x d c. S S! R x x T x x c Th Fourir sris o whr d x o th itrvl LL, is 0 x x x cos bsi L L L x x L L L L cos, 0,,, 3, x b x L L L si,,, 3, rity: g x 0 0 i g x is odd g x i g x is v which ld to hl-rg Fourir sris or x o th itrvl 0, L.

10 ENGI 344 Appdix Formulæ g A-0 Sris solutios o ODEs: Assum tht y x with is solutio o px qx y r x 0 y 0 (Mcluri sris)! d y Eithr us th ODE to id y 0 r 0 p0 y0 q0 y0 d dirtit y x r x px yx qx yx y to id th highr drivtivs y x d hc 0 or Substitut y x id th vlus o th 0 roduct rul or highr drivtivs: k k uv Ck k k k 0 d d u d v succssivly! x ito th ODE d mtch coicits o d u d u dv d u d v v 3 3 d u d v d v u x to END OF AENDIX