MATH 308: Diff Eqs, BDP10 EXAMPLES [Belmonte, 2019] 1 Introduction 1.1 Basic Mathematical Models; Direction Fields

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1 MATH 308: Diff Eqs, BDP0 EXAMPLES Blmon, 09 Mos problms ar from NSS9 Inroducion Basic Mahmaical Modls; Dircion Filds / Plo a dircion fild for dy/dx = 4x/y (a) Vrify ha h sraigh lins y = ±x ar soluions curvs, providd x 0 Plo hm (b) Plo h soluion curv wih iniial condiion y(0) = (c) Plo h soluion curv wih iniial condiion y() = (d) Wha can you say abou h bhavior of h soluions in (b) and (c) as x +? How abou as x? /3 A modl for h vlociy v a im of a crain objc falling undr h influnc of graviy in a viscous mdium is givn by h quaion dv/d = v/8 (a) Plo a dircion fild for h diffrnial quaion (b) Plo h soluions wih h iniial condiions v(0) = 5,8,5 (c) Why is h valu v = 8 calld h rminal vlociy? 3 /4 If h viscous forc in /3 is nonlinar, a possibl modl would b providd by h diffrnial quaion dv/d = v 3 /8 (a) Plo a dircion fild for h diffrnial quaion (b) Plo h soluions wih h iniial condiions v(0) = 0,,,3 (c) Wha is h rminal vlociy in his cas? 4 /5 Th logisic quaion for h populaion (in housands) of a crain spcis is givn by d p/d = 3p p (a) Plo a dircion fild for h diffrnial quaion (b) If h iniial populaion is 3000 i, p(0) = 3, wha can you say abou h limiing populaion lim p()? (c) If p(0) = 08, wha is lim p()? (d) Can a populaion of 000 vr dclin o 800? 5 /6 Considr h diffrnial quaion dy/dx = x + sin y (a) A soluion curv passs hrough h poin (,π/) Wha is is slop a his poin? (b) Argu ha vry soluion curv is incrasing for x > (c) Show ha h scond drivaiv of vry soluion saisfis d y dx = + xcosy + siny (d) A soluion curv passs hrough (0,0) Prov ha his curv has a rlaiv minimum a (0,0) 6 /7 Considr h quaion d p/d = p(p )(p ) for h populaion (in housands) of a crain spcis a im (a) Plo a dircion fild for h diffrnial quaion (b) If h iniial populaion is 4000 i, p(0) = 4, wha can you say abou h limiing populaion lim p()? (c) If p(0) = 7, wha is lim p()? (d) If p(0) = 08, wha is lim p()? () Can a populaion of 900 vr incras o 00? 7 /0 For ach of h following diffrnial quaions, plo h dircion fild along wih som soluion curvs (a) dy/dx = sinx (b) dy/dx = siny (c) dy/dx = sinxsiny (d) dy/dx = x + y () dy/dx = x y 8 / Plo h dircion fild for dy/dx = x/y along wih h soluion curv saisfying y(0) = 4 9 /5 Plo h dircion fild for dy/dx = x y along wih h soluion curv saisfying y(0) = 0 0 3/7 Plo h dircion fild for dy/dx = 3 y + x Wha can you say abou h bhavior of soluions as x +? Soluions of Som Diffrnial Equaions 3/a Show ha φ (x) = x is an xplici soluion o h diffrnial quaion x dy dx = y on h inrval (,) 3/a Show ha y + x 3 = 0 is an implici soluion o h diffrnial quaion dy dx = /(y) on h inrval (,3) 3 3/3 Drmin whhr y = x + sinx is a soluion of h diffrnial quaion d y dx + y = x + 4 4/7 Drmin whhr y = x 3 x is a soluion of h diffrnial quaion d y dy dx dx y = 0 5 4/9 Drmin whhr x + y = 4 is an implici soluion of h diffrnial quaion dy/dx = x/y 6 4/ Drmin whhr xy + y = x is an implici soluion of h diffrnial quaion dy dx = xy y xy + x 7 4/4 Show ha φ (x) = c sinx + c cosx is a soluion o d y dx + y = 0 for any choic of consans c and c Hnc φ (x) is a wo-paramr family of soluions 8 4/7 Show ha φ (x) = +C 3x is a soluion o dy/dx 3y = 3 for any choic of consan C Hnc φ (x) is a on-paramr family of soluions Plo som soluions

2 9 4/0a For which valus of m is φ (x) = mx a soluion of d y + 6 dy dx dx + 5y = 0? 0 4/a For which valus of m is φ (x) = x m a soluion of 3x d y + x dy dx dx 3y = 0? 3 Classificaion of Diffrnial Equaions In # 8, classify h diffrnial quaion according o h following criria Th filds in which hy aris ar mniond ODE or PDE (ordinary or parial diffrnial quaion) ordr of h diffrnial quaion indpndn and dpndn variabls linar or nonlinar (for ODEs only) 5/ 5 d x + 4 dx d d + 9x = cos3 (mchanical vibraions, lcric circuis, sismology) 5/3 dy dx = y( 3x) x( 3y) (compiion bwn spcis, cology) 3 5/4 u + u = 0 (Laplac s quaion, ponial hory, x y lcriciy, ha, arodynamics) ( ( ) ) 4 5/5 y + dy dx = C (calculus of variaions) 5 5/6 dx d = k (4 x)( x), wih k consan (racion ras) 6 5/7 d p d = kp(p p), whr k and P ar consans (logisic curv, pidmiology, conomics) 7 5/9 x d y + dy dx dx + xy = 0 (arodynamics, srss analysis) 8 5/ N = N + r r N r + kn, k consan (nuclar fission) 9 5/3 Th ra of chang of h populaion p of bacria a im is proporional o h populaion a im Wri a diffrnial quaion ha fis his physical dscipion 0 5/5 Th ra of chang in coff mpraur Q a im is proporional o h diffrnc bwn air mpraur M a im and coff mpraur Q a im Wri a diffrnial quaion ha fis his physical dscipion Firs-Ordr Diffrnial Eqs Linar Equaions; Mhod of Ingraing Facors 54/7 Obain h gnral soluion o dy dx y 3x = 0 54/8 Solv dy dx = y x + x /9 Solv dθ dr + r anθ = scθ 4 54/ Solv ( + y + ) d dy = / Solv dy dx = x 4x 4y 6 54/3 Solv y dx dy + x = 5y3 7 54/4 Solv x dy dx + 3( y + x ) = sinx x 8 54/5 Solv ( x + ) dy dx + xy x = /7 Solv h iniial valu problm dy dx y x = xx, y() = 0 54/9 Solv h iniial valu problm dx d + 3x = +4 ln, x() = 0 Sparabl Diffrnial Eqs 46/ Drmin whhr dy dx sin(x + y) = 0 is sparabl 46/5 Drmin if ( xy + 3y ) dy xdx is sparabl 3 46/7 Solv h quaion x dy dx = y /9 Solv h quaion dx d = x +x 5 46/5 Solv h quaion ( x + xy ) dx + x ydy = /7 Solv h iniial valu problm (IVP) dy dx = ( + y ) anx, y(0) = / Solv h IVP θ dy dθ = ysinθ y +, y(π) = 8 46/3 Solv h IVP dy d = cos y, y(0) = π/4 9 47/33 Mixing A ank is iniially filld wih 400L of war conaining kg of sal Brin (a sal soluion) conaining 03kg/L nrs h ank a 0L/min Th wll-sirrd mixur flows ou of h ank a h sam ra Find h mass of sal in h ank afr 0min 0 47/37 Compound Inrs If P() is h amoun of dollars in a savings bank accoun ha pays a yarly inrs ra of r% compoundd couninuously, hn dp d = 00 r P, wih in yars Assum h inrs is 5% annually, P(0) = $000, and no monis ar wihdrawn (a) How much will b in h accoun afr yr? (b) Whn will h accoun rach $4000? (c) If $000 is addd o h accoun vry monhs, how much will b in h accoun afr 35yr?

3 3 Modling wih s-ordr Eqs 00/ Mixing A ank conains 00L of war wih 05kg of sal A sal soluion conaining 005kg/L nrs a 8L/min Th mixur flows ou of h ank a h sam ra Find h mass of sal in h ank afr min Whn will h ank s sal concnraion rach 0kg/L? 00/9 In 990 h Dparmn of Naural Rsourcs rlasd 000 splak (a crossbrd of fish) ino a lak In 997 h populaion of splak in h lak was 3000 Using h xponnial law of populaion growh, drmin h populaion of splak in h lak in h yar /4 If iniially hr ar 300g of a radioaciv subsanc and afr 5yr hr ar 00g rmaining, how much im mus laps bfor only 0g rmain? 4 07/ A cup of ho coff inially a 95 C cools o 80 C in 5min in a room of mpraur C Us Nwon s Law of Cooling o drmin whn h mpraur of h coff will b a nic 50 C and rady o drink 5 07/3 A rd win a room mpraur 70 F is chilld in ic (3 F) If i aks 5min for h win o chill o 60 F, how long will i ak for h win o rach 56 F, rady o drink? 6 08/7 On a ho Saurday morning whil popl ar working insid, h air condiionr kps h mpraur insid h building a 4 C A noon h air condiionr is urnd off, and h popl go hom Th mpraur ousid is 35 C for h rs of h afrnoon (a) If h im consan for h building is 4hr, wha will b h mpraur insid h building a :00 pm? (b) A 6:00 pm? (c) Whn i rach 7 C? 7 5/ An objc of mass 5kg is rlasd from rs 000m abov h ground I falls undr h influnc of graviy Th forc du o air rsisanc is proporional o h vlociy of h objc wih proporionaliy consan b = 50N-sc/m Drmin h quaion moion for h objc Whn will h objc hi h ground? 8 5/7 A parachuis whos mass (wih quipmn) is 75kg drops from a hlicopr hovring 000m abov h ground and falls oward h ground undr h influnc of graviy Th forc du o air rsisanc is proporional o h vlociy of h parachuis, wih consans b = 30N-sc/m whn h chu is closd and b = 90N-sc/m whn h chu is opn If h chu opns whn h vlociy of h parachuis his 0m/s, afr how many sconds will sh rach h ground? 9 / An RL circui wih a 5-Ω rsisor and a 005-H inducor carris a currn of A a = 0, a which im a volag sourc E () = 5 cos 0 V is addd Drmin subsqun inducor currn and volag 0 /7 An indusrial lcromagn can b modld as an RL circui, whil i is bing nrgizd wih a volag sourc If h inducanc is 0H and h wir windings conain 3Ω of rsisanc, how long dos i ak a consan applid volag o nrgiz h lcromagn o wihin 90% of is final valu (i, h currn quals 90% of is asympoic valu)? 4 Diffrncs Bwn Linar & Nonlinar Diffrnial Eqs 76/ Us a horm o drmin h largs opn inrval in which h soluion of h iniial valu problm ( 4)y + y = 0, y() =, is crain o xis 76/4 Sam as # for ( 4 ) y + y = 3, y( 3) = 3 76/6 Sam as # for h IVP (ln)y + y = co, y() = /8 Us a horm o drimin h largs opn s D in h y-plan in which a soluion of h iniial valu problm y = y, y( 0 ) = y 0, xiss ovr som opn inrval 0 h < < 0 + h abou 0 Hr ( 0,y 0 ) D 5 76/0 Sam as #4 for y = ( + y ) 3/, y(0 ) = y / Sam as #4 for dy d = yco + y, y( 0) = y /4 Solv h IVP y = y, y(0) = y 0 and drmin how h inrval in which h soluion xiss dpnds on y /6 Sam as #7 for y = ( + 3 )y, y(0) = y /8 Plo h dircion fild and svral soluion curvs for h diffrnial quaion y = y(3 y) Dscrib how soluions apprar o bhav as incrass and how hir bhavior dpnds on h iniial valu y 0 whn = /0 Sam as #9 for y = y 5 Auonomous Diffrnial Eqs & Populaion Dynamics 88/ Drmin and classify h quilibrium soluions of dy/d = ay + by, a > 0, b > 0, y 0 R Plo a dircion fild and skch svral soluions in h y-plan 88/4 Sam as # for dy/d = y, y 0 R 3 88/6 Sam as # for dy d = arcany + y, y 0 R 4 89/8 Sam as # for dy/d = k (y ), k > 0, y 0 R 5 89/0 Sam as # for dy/d = y ( y ), y 0 R 6 89/ Sam as # for dy/d = y ( 4 y ), y 0 R 7 89/3 Sam as # for dy/d = y ( y ), y 0 R 3

4 8 90/6 Anohr quaion usd o modl populaion growh is h Gomprz quaion dy/d = ryln(k/y), whr r and K ar posiiv consans (a) Drmin and classify is quilibrium soluions (b) For 0 y K, drmin whr h graph of y vrsus is concav up and whr i is concav down 9 90/7 Rcall h Gomprz quaion from #8 (a) Solv h i subjc o h iniial condiion y(0) = y 0 (b) Find y() if r = 07, K = , and y 0 /K = 05 (c) Wih h sam daa as in (b), find h im τ a which y(τ) = 075K 0 94/8 In chmical racions, h diffrnial quaion dx/d = α (p x)(q x) ariss, whr α, p, q ar posiiv consans (a) If p q and x(0) = 0, drmin lim x() wihou solving h quaion Thn solv h quaion o vrify your assrion (b) If p = q and x(0) = 0, drmin lim x() wihou solving h quaion Thn solv h quaion o vrify your assrion 6 Exac Diffrnial Equaions and Ingraing Facors 64/9 Drmin if h (xy + 3) dx + ( x ) dy = 0 is xac If i is, hn solv i 64/ Sam as # for h diffrnial quaion ( x siny 3x ) dx + ( x cosy + 3 y /3) dy = /3 Sam as # for (y )d + ( + )dy = /5 Sam as # for cosθ dr ( r sinθ θ ) dθ = 0 ( ) ( ) 5 64/9 Sam for x + y dx + x y dy = 0 +x y +x y 6 64/ Solv ( x + y x ) dx + ( yx cosy ) dy = 0, y() = π 7 64/3 Solv ( y + y)d + ( + ) dy = 0, y(0) = 8 69/7 Solv h quaion (xy)dx + ( y 3x ) dy = /9 Solv h quaion ( x 4 x + y ) dx xdy = /3 Find an ingraing facor of h form x n y m and solv h quaion ( y 6xy ) dx + ( 3xy 4x ) dy = 0 3 Scond-Ordr Linar Diffrnial Equaions 3 Homognous Diffrnial Eqs wih Consan Coffs 64/ Find a gnral soluion o y + 7y 4y = 0 64/5 Find a gnral soluion o y + 8y + 6y = /5 Solv y 4y + 3y = 0, y(0) =, y (0) = /7 Solv y 6y + 9y = 0, y(0) =, y (0) = /9 Solv y + y + y = 0, y(0) =, y (0) = / Find a gnral soluion o 3y 7y = /3 Find a gnral soluion o 5y + 4y = /4 Find a gnral soluion o 3z + z = /7 Drmin if y () = cos sin and y () = sin ar linarly dpndn on (0, ) Us h Wronskian from /9 Drmin if y () = and y () = ar linarly dpndn on (0, ) Us h Wronskian from 3 3 Soluions of Linar Homognous Equaions; h Wronskian 55/ Find h Wronskian for h funcion pair cos, sin 55/4 Sam as # for x, x x 3 55/6 Sam as # for cos θ, + cosθ 4 55/8 Drmin h largs inrval in which h IVP ( )y 3y + 4y = sin, y( ) =, y ( ) =, is crain o hav a uniqu wic-diffrniabl soluion 5 55/0 Sam as #4 for y + (cos)y + 3(ln )y = 0, y() = 3, y () = 6 56/ Sam as #4 for (x )y + y + (x )(anx)y = 0, y(3) =, y (3) = 7 56/4 Vrify ha y () = and y () = / ar soluions of h diffrnial quaion yy + (y ) = 0 for > 0 Thn show ha y = c + c / is no, in gnral, a soluion of his quaion Dos his conradic h Principl of Suprposiion? 8 56/8 If h Wronskian of f and g is W =, and if f () =, find g() 9 56/0 If h Wronskian of f and g is W = cos sin, and if u = f + 3g and v = f g, hn find h Wronskian of u and v 0 56/ Find h fundamnal s of soluions y, y, of h diffrnial quaion y + y y = 0 ha saisfy h iniial condiions y (0) =, y (0) = 0 and y (0) = 0, y (0) = 4

5 33 Complx Roos of h Characrisic Equaion 7/ Find a gnral soluion o y + 9y = 0 7/3 Find a gnral soluion o z 6z + 0z = 0 3 7/9 Find a gnral soluion o y 8y + 7y = 0 4 7/ Find a gnral soluion o z + 0z + 5z = 0 5 7/3 Find a gnral soluion o y y + 6y = 0 6 7/5 Find a gnral soluion o y 3y y = 0 7 7/ Solv y + y + y = 0, y(0) =, y (0) = 8 7/5 Solv y y + y = 0, y(π) = π, y (π) = 0 9 7/8 To s h ffc of changing h paramr b in h IVP y + by + 4y = 0, y(0) =, y (0) = 0, solv h problm for b = 5,4, and plo h soluions 0 73/36 Is is possibl o us h form d (α+βi) + d (α βi) o solv IVPs whr h cofficins d and d ar complx consans (a) Us his form o solv #7 abov (b) Show ha, in gnral, d and d mus b complx conjugas in ordr for h soluion o b ral-valud 34 Rpad Roos; Rducion of Ordr 7/ Find h gnral soluion of 9y + 6y + y = 0 7/4 Sam as # for 4y + y + 9y = 0 3 7/6 Sam as # for y 6y + 9y = /8 Sam as # for 6y + 4y + 9y = /0 Sam as # for y + y + y = / Solv h iniial valu problm y 6y + 9y = 0, y(0) = 0, y (0) = Plo h soluion and dscrib is bhavior for incrasing 7 73/4 Sam as #6 for y + 4y + 4y = 0, y( ) =, y ( ) = 8 73/6 Find h soluion y() of y y + 4 y = 0, y(0) =, y (0) = b as a funcion of b Drmin h criical valu of b ha sparas soluions ha grow posiivly from hos ha vnually grow ngaivly Illusra 9 73/8 Considr h IVP 9y + y + 4y = 0, y(0) = a > 0, y (0) = (a) Solv for y() as a funcion of a (b) Find h criical valu of a ha sparas h soluions ha bcom ngaiv from hos ha ar always posiiv 0 74/4 Givn h soluion y () = o h diffrnial quaion y + y y = 0, us Rducion of Ordr o find a scond soluion 35 Nonhomognous Equaions: Mhod of Undrmind Cofficins 80/ Dcid whhr h Mhod of Undrmind Cofficins (MUC) can b applid o find a paricular soluion of h diffrnial quaion y + y y = 80/3 Dcid if MUC may b usd o find a soluion of h diffrnial quaion y 6y + y = (sinx)/ 4x 3 80/3 Us MUC o find a paricular soluion of h diffrnial quaion y y + 9y = 3sin3 4 80/5 Sam as #3 for d y dx 5 dy dx + 6y = xx 5 80/ Sam as #3 for x 4x + 4x = 6 80/7 Sam as #3 for y + 9y = 4 3 sin3 7 86/7 Find a gnral soluion o y y 3y = /9 Sam as #7 for y 3y + y = x sinx 9 86/7 Solv h IVP y y y = cosx sinx; y(0) = 7 0, y (0) = /9 Sam as #9 for y y = sinθ θ ; y(0) =, y (0) = 36 Variaion of Paramrs 9/ Us Variaion of Paramrs (VOP) o find a gnral soluion o y + 4y = an 9/3 Us VOP o solv y y + y = 3 9/5 Sam as #3 for y + 6y = sc4θ 4 9/7 Sam as #3 for y + 4y + 4y = ln 5 9/9 Us boh MUC and VOP o solv y y = / Sam as #3 for y + y = an /3 Sam as #3 for v + 4v = sc 4 () 8 9/5 Sam as #3 for y + y = 3sc + 9 9/7 Sam as #3 for y + y = an 0 9/3 Show ha y = and y = + ar soluions of y ( + )y + y = 0 Us VOP o find a gnral soluion of y ( + )y + y = 5

6 37 Mchanical and Elcrical Vibraions 0/ A kg mass is aachd o a spring wih siffnss k = 50N/m Th mass is displacd 4 m o h lf of h quilibrium poin and givn a vlociy of m/s o h lf Nglcing damping, find h quaion of moion of h mass along wih h ampliud, priod, and frquncy Whn dos h mass firs pass hrough h quilibrium posiion? 0/ A 3kg mass is aachd o a spring wih siffnss k = 48N/m Th mass is displacd m o h lf of h quilibrium poin and givn a vlociy of m/s o h righ Th damping forc is ngligibl Find h quaion of moion of h mass along wih h ampliud, priod, and frquncy Whn dos h mass firs cross h quilibrium posiion? 3 0/3 Th moion of a mass-spring sysm wih damping is govrnd by y + by + 6y = 0, y(0) =, y (0) = 0 Find h quaion of moion and plo is graph for b = 0,6,8,0 4 0/4 Th moion of a mass-spring sysm wih damping is govrnd by y + by + 64y = 0, y(0) =, y (0) = 0 Find h quaion of moion and plo is graph for b = 0,0,6,0 5 0/5 Th moion of a mass-spring sysm wih damping is govrnd by y + 0y + ky = 0, y(0) =, y (0) = 0 Find h quaion of moion and plo is graph for k = 0,5,30 6 0/6 Th moion of a mass-spring sysm wih damping is govrnd by y + 4y + ky = 0, y(0) =, y (0) = 0 Find h quaion of moion and plo is graph for k =,4,6 7 0/7 A 8 kg mass is aachd o a spring wih siffnss 6N/m Th damping consan for h sysm is N-sc/m If h mass if movd 3 4 m o h lf of quilibrium and givn an iniial lfward vlociy of m/s, drmin h quaion of moion of h mass and giv is damping facor, quasipriod, and quasifrquncy 8 0/9 A kg mass is aachd o a spring wih siffnss 40N/m Th damping consan is 8 5N-sc/m If h mass if movd 0 m o h righ of quilibrium and givn an iniial righward vlociy of m/s, find h maximum displacmn from quilibrium ha i will aain 9 0/ A kg mass is aachd o a spring wih siffnss 00N/m Th sysm s damping consan is 5 N-sc/m If h mass if pushd righward from h quilibrium posiion wih a vlociy of m/s, whn will i aain is maximum displacmn o h righ? 0 / A 4 kg mass is aachd o a spring wih siffnss 8N/m Th damping consan for h sysm is N-sc/m If h mass if pushd m o h lf of quilibrium and givn a lfward vlociy of m/s, whn will h mass aain is maximum displacmn o h lf? 38 Forcd Priodic Vibraions 7/3 Find h quaion of moion for an undampd sysm y + 9y = cos3; y(0) =, y (0) = 0 Plo h soluion 8/4 Sam as # for y + y = 5cos; y(0) = 0, y (0) = 3 8/5 Solv my + ky = F 0 cosγ ; y(0) = 0, y (0) = 0, wih γ ω = k/m Plo h soluion for F 0 = 3, m =, ω = 9 and γ = 7 Th phnomnon is rfrrd o as bas and is usd in uning sringd insrumns Th sam phnomnon in lcronics is calld ampliud modulaion, as in AM radio 4 8/8 Th rspons of an ovrdampd sysm o a consan forc is govrnd by h quaion my + by + ky = F 0 cosγ wih m =, b = 8, k = 6, F 0 = 8, and γ = 0 If h sysm sars from rs y(0) = y (0) = 0, compu and plo h displacmn y() Wha is h limi of y() as +? Inrpr his physically 5 8/9 An 8kg mass is aachd o a spring hanging from h ciling, hrby causing h spring o srch 96m upon coming o rs a quilibrium Th damping consan for h sysm is 3N-sc/m A im = 0, a forc F () = cos N is applid Drmin h sady-sa soluion for h sysm 6 8/0 Show ha h priod of h simpl harmonic moion of a mass hanging from a spring is π l/g whr l dnos h amoun (byond is naural lngh) h spring is srchd whn h mass is a quilibrium 7 8/ A mass wighing 8lb is aachd o a spring hanging from h ciling and coms o rs a is quilibrium posiion A = 0, an xrnal form of F () = cos lb is applid o h sysm If h spring consan is 0lb/f and h damping consan is lb-sc/f, find h quaion of moion of h mass Wha is h rsonanc frquncy for h sysm? 8 8/3 A mass wighing 3lb is aachd o a spring haning from h ciling and coms o rs a is quilibrium posiion A = 0, an xrnal form of F () = 3cos4 lb is applid o h sysm If h spring consan is 5lb/f and h damping consan is lb-sc/f, find h sady-sa soluion for h sysm 9 8/4 An 8kg mass is aachd o a spring hanging from h ciling and allowd o com o rs Th spring consan is 40N/m and h damping consan is 3N-sc/m A im = 0, an xrnal forc of sin( + π/4) is applid Find h ampliud and frquncy of h sady-sa soluion 0 8/5 An 8kg mass is aachd o a spring hanging from h ciling and allowd o com o rs Th spring consan is 40N/m and h damping consan is 3N-sc/m A im = 0, an xrnal forc of sin4 N is applid Drmin h ampliud and frquncy of h sady-sa soluion 5 Sris Soluions of Scond-Ordr Linar Eqs 5 Rviw of Powr Sris 433/ Drmin h radius and inrval of convrgnc for h powr sris n=0 n n + (x )n 6

7 433/3 Sam as # for 3 433/5 Sam as # for n n=0 n= n (x + )n 3 n 3 (x )n 4 434/8 Drmin radii of convrgnc for hs sris (a) k=0 k x k (b) (c) Maclaurin sin sris () sinx x = n=0 ( ) n x n (n + )! k=0 k+ x k+ (d) Maclaurin cosin sris (f) k=0 k x 4k 5 434/7 Find a powr sris xpansion for f (x) givn h xpansion f (x) = ( + x) = n=0 ( ) n x n 6 434/ Find a powr sris xpansion for g(x) = x f () d givn h xpansion f (x) = ( + x) = 7 435/7 Show ha x n=0 n(n + )a n x n = n= by shifing h summaion indx n=0 0 ( ) n x n (n )(n )a n x n 8 435/9 Giv h Taylor sris abou x 0 = π for f (x) = cosx 9 435/3 Giv h Maclaurin sris for f (x) = + x x 0 435/3 Giv h Maclaurin sris for f (x) = ln( + x) 5 Sris Soluions Nar an Ordinary Poin, Par I 443/3 Drmin all singular poins of h diffrnial quaion ( θ ) y + y + (sinθ)y = 0 443/5 Sam for ( ) x + ( + )x ( )x = /9 Sam for (sinθ)y (lnθ)y = / Find a las h firs 4 nonzro rms in a powr sris xpansion abou x = 0 for a gnral soluion o h diffrnial quaion y + (x + )y = /5 Sam as #4 for y + (x )y + y = /7 Sam as #4 for w x w + w = / Find a powr sris xpansion abou x = 0 for a gnral soluion o h diffrnial quaion y xy + 4y = 0 Includ a gnral formula for h cofficins 8 444/3 Sam as #7 for z x z xz = /5 Find a las h firs four nonzro rms in a powr sris xpansion abou x = 0 for h soluion o h IVP w + 3xw w = 0; w(0) =, w (0) = /9 Find a cubic polynomial approximaion for h soluion o h IVP y + y xy = 0; y(0) =, y (0) = Plo linar, quadraic, and cubic polynomial approximaions for 5 x 5 53 Sris Soluions Nar an Ordinary Poin, Par II 449/ Find a mimimum valu for h radius of convrgnc of a sris soluion of (x + )y 3xy + y = 0 abou x 0 = 449/5 Sam as # for y (anx)y + y = 0 abou x 0 = /7 Find a las h firs four nonzro rms in a sris xpansion abou x = for a soluion o y + (x )y = / Sam as #3 for x y y + y = 0 abou x 0 = 5 449/3 Find a las h firs four nonzro rms in a powr sris xpansion of h soluion o h iniial valu problm x + (sin )x = 0; x(0) = 6 449/7 Sam for y (sinx)y = 0; y(π) =, y (π) = /9 Sam for y x y + (cosx)y = 0; y(0) =, y (0) = 8 450/ Find a las h firs four nonzro rms in a powr sris xpansion abou x = 0 of a gnral soluion o h diffrnial quaion y xy = sinx 9 450/3 Sam as #8 for z + xz + z = x + x /7 Sam as #8 for ( x ) y y + y = anx 6 Th Laplac Transform 6 Dfiniion of h Laplac Transform 360/7 Us h dfiniion o compu h Laplac ransform of f () = cos3 360/3 Us a Laplac ransform abl plus linariy o find h Laplac ransform of f () = /5 Sam as # for f () = cos {, 0, 4 360/ Drmin whhr f () = ( ) < 0 is coninuous, picwis coninuous, or nihr on 0,0 Plo h graph of f (), 0 <, 5 360/3 Sam as #4 for f () = < < 3, 4 3 < /5 Sam as #4 for f () =

8 7 365/ Find L { f ()} for f () = + sin using abls, linariy, or propris of h Laplac ransform 8 365/7 Sam as #7 for f () = ( ) / Sam as #7 for f () = coshb 0 365/5 Sam as #7 for f () = cos 3 374/ Find invrs Laplac ransform of F (s) = 6/(s ) 4 390/5 Exprss h funcion blow using sp funcions Thn compu is Laplac ransform 0, 0 < <,, < <, g() = < < 3, 3 3 < 3 390/7 Sam as # for h funcion dpicd blow 374/9 Sam as # for F (s) = 3s 5 s 4s / Sam as # for F (s) = 6s 3s + s(s )(s 6) 4 375/3 Sam as # for F (s) = 5s + 34s + 53 (s + 3) (s + ) 5 375/5 Sam as # for F (s) = 7s + 3s + 30 (s )(s + s + 5) 4 390/9 Sam as # for h funcion dpicd blow 6 375/7 Sam as # givn s F (s) 4F (s) = 5 s + 6 Soluion of IVPs via Laplac Transforms 38/ Solv y y + 5y = 0; y(0) =, y (0) = 4 38/3 Solv y + 6y + 9y = 0; y(0) =, y (0) = /5 Solv w + w = + ; w(0) =, w (0) = 4 38/7 Solv y 7y + 0y = 9cos + 7sin; y(0) = 5, y (0) = /5 Solv y 3y + y = cos; y(0) = 0, y (0) = 5 390/3 Find h invrs Laplac ransform of F (s) = ( s 3 4s) /(s + ) 6 390/7 Sam as #5 for F (s) = 3s (s 5) (s + )(s + ) 7 396/3 Compu{ h Laplac ransform of h priodic, 0 < <, funcion f () = whr f () has priod, < <, 8 396/5 Sam as #7 for h funcion dpicd blow 6 38/7 Solv y + y y = 3 ; y(0) =, y (0) = /9 Solv y + 5y y = ; y(0) =, y (0) = 8 38/ Solv y y + y = cos sin; y(0) =, y (0) = /9 Solv y 4y + 3y = 0; y(0) = a, y (0) = b 0 383/3 Solv y + y + y = 5; y(0) = a, y (0) = b 9 396/7 Sam as #7 for h funcion dpicd blow 63 Sp/Priodic Funcions 390/ Plo h graph of f () = ( ) u( ), whr u is h Havisid uni sp funcion Thn compu h Laplac ransform of f () 8

9 0 396/8 Sam as #7 for h funcion dpicd blow 64 Diffrnial Equaions wih Disconinuous Forcing Funcs 390/9 Th currn I () in an RLC sris circui is modld by I () + I () + I () = g(); I (0) = 0, I (0) = 0, whr 0, 0 < < 3π, g() = 0, 3π < < 4π, 0, 4π < Drmin h currn as a funcion of Plo I () for 0 < < 8π 39/ Solv y + y = u( 3); y(0) = 0, y (0) = 3 39/3 Solv y + y = ( 4)u( ), y(0) = 0, y (0) = 4 39/5 Solv y + y + y = u( π) u( 4π), y(0) =, y (0) = 5 39/7 Solv z + 3z + z = 3 u( ); z(0) =, z (0) = /9 Solv{ y + 4y = g(); y(0) =, y (0) = 3, sin, 0 π, whr g() = 0, π < 7 39/3 Solv y + 5y + 6y = g(); y(0) = 0, y (0) =, 0, 0 <, whr g() =, < 5, 5 < 8 39/3 Solv{ y + 3y + y = g(); y(0) =, y (0) =,, 0 < 3, whr g() = 3 < 9 39/33 Th mixing ank shown blow iniially holds 500L of a brin soluion wih a sal concnraion of 00kg/L For h firs 0min, valv A is opn, adding L/min of brin conaining a 004kg/L sal concnraion Afr 0min, valv B is swichd in, adding a 006kg/L sal concnraion a L/min Th xi valv C rmovs L/min, hrby kping h volum consan Find h concnraion of sal in h ank as a funcion of im Plo i 0 39/34 Suppos in #9 valv B is iniially opnd for 0min and hn valv A is swichd in for 0min Finally, valv B is swichd back in Find h concnraion of sal in h ank as a funcion of im Plo i 65 Impuls Funcions 40/3 Evalua ( ) δ () d whr δ is h Dirac dla funcion 40/7 Drmin h Laplac ransform of f () = δ ( ) δ ( 3) 3 40/9 Sam as # for f () = δ ( ) 4 40/ Sam as # for f () = δ ( π)sin 5 40/3 Solv h symbolic IVP w + w = δ ( π); w(0) = 0, w (0) = /5 Sam as #5 for y + y 3y = δ ( ) δ ( ); y(0) =, y (0) = 7 40/7 Sam as #5 for y y = 4δ ( ) + ; y(0) = 0, y (0) = 8 40/9 Sam as #5 for w + 6w + 5w = δ ( ); w(0) = 0, w (0) = / Sam as #5 for y + y = δ ( π); y(0) = 0, y (0) = Also plo h soluion 0 40/3 Sam as #9 for y + y = δ ( π) + δ ( π); y(0) = 0, y (0) = 9

10 66 Th Convoluion Ingral 404/ Us h Convoluion Thorm o obain a formula for h soluion o h IVP y y + y = g(); y(0) = ; y (0) =, whr g() is picwis coninuous on 0,) and of xponnial ordr 404/3 Sam for y + 4y + 5y = g(), y(0) = ; y (0) = 3 404/5 Us h Convoluion Thorm o find h invrs Laplac ransform of F (s) = s(s + ) 4 404/7 Sam as #3 for F (s) = 5 404/5 Solv h ingral quaion y() y(v)sin( v) dv = 4 (s + )(s 5) 6 404/7 Sam as #5 for y() + 0 ( v)y(v) dv = 7 404/9 Sam as #5 for y() + 0 ( v) y(v) dv = / Solv h ingro-diffrnial quaion y (5) + y() 0 y(v)sin( v) dv = sin, y(0) = 9 404/3 A linar sysm is govrnd by y + 9y = g(); y(0) =, y (0) = 3 (a) Find h ransfr funcion H (s) for h sysm (b) Find h impuls rspons funcion h() (c) Giv a formula for h soluion of h IVP 0 404/7 Sam as #9 for y y + 5y = g(); y(0) = 0, y (0) = 7 Sysms of Firs-Ordr Linar Equaions 7 Inroducion o Sysms of s-ordr Linar Equaions 500/ Pu h sysm in marix form: 500/3 Sam as # for 3 500/5 Sam as # for x = x + y + z y = z x z = 4y x = 7x + y y = 3x y x = (sin)x + y y = (cos)x + ( a + b 3) y 4 500/7 Exprss h highr ordr diffrnial quaion my + by + ky = 0 (dampd mass-spring oscillaor) as a marix sysm in normal form 5 500/9 Sam as #4 for y y = 0 (h Airy quaion) 6 500/ Sam as #4 for h highr ordr sysm x + 3x + y = 0, y x = /3 Sam as #4 for h highr ordr sysm x 3x + y (cos)x = 0, y + y x + y + x = 0 7 Marics 0 53/ L A = and B = Find (a) A + B and (b) 3A B /3 L A = and B = 5 Find (a) AB, (b) A = AA, and (c) B = BB /5 L A =, B =, and 3 C = Find (a) AB, (b) AC, and (c) A(B + C) 4 53/9 Compu h invrs of h marix A = if i xiss 5 53/3 Sam as #4 for A = /7 Sam as #4 for X() = /9 Sam as #4 for X() = 8 54/ Evalua h drminan 9 54/5 Sam as #8 for /7 Drmin h valus of r for which d(a ri) = 0, whr I is h idniy marix and A = 4 54/3 Find dx/d for x = /33 Find dx/d for X = /35 Vrify ha x() = 3 x = 4 x 4 54/37 Vrify ha X() = sysm X = X 4 3 solvs h sysm 3 3 solvs h 0

11 5 54/39 For A() = cos sin & B() = sin cos find (a) A() d, (b) 0 B() d, and (c) d d (A()B()) 73 Sysms of Linar Algbraic Eqs; Linar Indpndnc, Eignvalus, Eignvcors 504/ Find all soluions o h sysm x + x + x 3 = 6 x + x + x 3 = 6 x + x + 3x 3 = 6 504/3 Sam as # for 3 504/5 Sam as # for 4 504/7 Sam as # for 5 504/9 Sam as # for x + x x 3 = 0 x x + x 3 = 0 x + x x 3 = 0 x + x = 0 x + 3x = 0 x + 3x = 0 3x + 9x = 0 ( i)x + x = 0 x ( + i)x = / Sam as # for x + x 3 = 3x + x + 4x 3 = x + x + 5x 3 = /3 Show ha h sysm blow has a uniqu soluion for r =, bu an infini numbr of soluions for r = Hr x 3x = rx, x x = rx 74 Basic Thory of Sysms of Firs-Ordr Linar Equaions 5/3 Wri h sysm blow in marix form x = Ax + f dx/d = x y z + dy/d = z + 5 dz/d = x y + 3z 5/5 Wri h quaion y () 3y () 0y() = sin as a firs-ordr sysm in normal form 3 5/7 Sam as # for d4 w dw 4 + w = 4 5/9 Wri h sysm x 5 0 = 4 as a s of scalar quaions 5 5/3 Drmin whhr x() = 3 6 5/5 Sam for x() = 5 x and y() = ar linarly dpndn or linarly indpndn on (,) and y() = 3 5, 7 5/ Th vcor funcions x = and ar soluions o a sysm x = Ax x = 4 Drmin if hy form a fundamnal soluion s If so, find a fundamnal marix for h sysm and giv a gnral soluion 8 5/3 Sam as #7 for x = x 3 = 3, x = 0, and 9 5/5 Vrify ha x = and x = ar 3 soluions on (,) of h homognous sysm x = Ax, whr A = and ha 3 x p = is a paricular soluion o h nonhomognous sysm x = Ax + f, whr f() = Find a gnral soluion o x = Ax + f 0 5/30 Vrify ha X() = 5 5 is a fundamnal 3 marix for h sysm x = Ax, whr A = Obain 3 h soluion o h IVP consising of h sysm plus h iniial condiion x(0) = by compuing his produc: x() = X()X (0)x(0) 3 75 Homognous Linar Sysms wih Consan Cofficins 53/ Find h ignvalus and ignvcors of 4 A = 53/3 Sam as # for A = /7 Sam as # for A = /9 Sam as # for A = / Find a gnral soluion of x () = Ax() for 3 A = /3 Sam as #5 for A =

12 7 53/9 Find a fundamnal marix for h sysm x () = Ax(), whr A = / Sam as #7 for A = /5 Us marix chniqus o find a gnral soluion of x = x + y + z h sysm y = x + z z = 4x 4y + 5z 0 53/3 Solv h IVP x = Ax, x(0) = x 0, 3 3 whr A =, x 3 0 = /3 Sam as #0 for A = 53/33 Sam as #0 for A = x 0 = 3 76 Complx Eignvalus 0, x 0 = 6 and 537/ Find a gnral soluion of h sysm x () = Ax() 4 for A = 5 537/ Sam as # for A = 3 537/3 Sam as # for A = 4 537/4 Sam as # for A = /5 Find a fundamnal marix for h sysm x () = Ax() for A = /6 Sam as #5 for A = /7 Sam as #5 for A = 8 537/ Sam as #5 for A = /3 Find h soluion o h sysm x () = Ax() for 3 A = for hs rspciv iniial condiions (a) x(0) = 0 (b) x(π) = (c) x( π) = 0 (d) x(π/) = 0 537/4 Sam as #3 for A = wih hs 0 iniial condiions (a) x(0) = (b) x( π) = 0 77 Fundamnal Marics 54/ Find a fundamnal marix for x = Ax, whr A = 4 54/4 Sam as # for A = 3 543/6 Sam as # for A = /0 Sam as # for A = /0 Sam as # for A = / Sam as # for A = /7 Sam as # for A = /8 Sam as # for A = /9 Sam as # for A = 0 55/0 Sam as # for A =

13 78 Rpad Eignvalus 55/ Drmin h xponnial marix A for 3 A = /3 Sam as # for A = /5 Sam as # for A = / Sam as # for A = / Sam as # for A = /7 Find a gnral soluion o h sysm x () = Ax(), whr A = /8 Sam as #6 for A = 8 55/0 Sam as #6 for A = / Solv h iniial valu problm x () = Ax(), x(0) =, whr A = / Sam as #9 for x(0) = A = Nonhomognous Linar Sysms and 54/ Us Variaion of Paramrs o find a gnral soluion 6 of h sysm x () = Ax() + f() wih A = and 4 3 f() = 5 54/3 Sam as # wih A = f() = /5 Sam as # wih A = f() = sin /7 Sam as # wih A = 0 f() = sin /9 Sam as # wih A = f() = sin 6 543/ Sam as # wih A = 7 543/3 Sam as # wih A = f() = and /5 Sam as # wih A = f() = /7 Sam as # wih A = f() = and and and and f() = 0 and and and 0 543/ Find h soluion o h sysm x () = Ax() + f() 0 for A = and f() = 3 for hs rspciv iniial condiions 5 (a) x(0) = 4 0 (b) x() = (c) x(5) = 0 4 (d) x( ) = 5 3

14 8 Numrical Mhods 8 Th Eulr or Tangn Lin Mhod 8/ Us Eulr s mhod o approxima h soluion o dy/dx = x/y, y(0) = 4, on 0, 05 using spsiz h = 0 8/3 Sam for dy/dx = x + y, y(0) = 3 8/5 Sam for y = x y, y() = 0, on,5 wih h = 0 4 8/7 Us Eulr s mhod o find approximaions o h soluion of h IVP y = siny, y(0) = 0, a x = π aking,, 4, and 8 sps 5 8/9 Us Eulr s mhod wih h = 0 o approxima h soluion o h IVP y = x y x y, y() = on h inrval x Compar hs soluions wih h acual soluion y = /x by graphing h xac and approxima soluions on h sam plo 6 8/4 Us Eulr s mhod wih h = 05,0,005,00 o approxima h soluion o h IVP y = xy, y(0) = on h inrval 0 x Graph h xac and approxima soluions on h sam plo 7 9/5 Us Eulr s mhod wih h = 3 o approxima h soluion o h IVP y = 004(93 y), y(0) = 360, a = 30 and = /6 Sam as #7 for h iniial valu problm y = ( 9 0 0)( 93 4 y 4), y(0) = Improvmns on h Eulr Mhod 30/7 Us h improvd Eulr s mhod wih h = 0 o approxima h soluion o y = x y, y() = 0, on,5 Plo h approxima soluion 30/8 Us h improvd Eulr s mhod wih h = 0 o approxima h soluion o y = ( y + y ) /x, y() = a x =,4,6,8 Plo 3 30/9 Us h improvd Eulr s mhod wih h = 0 o approxima h soluion o y = x + 3cos(x + y), y(0) = 0 a x = 0,04,,0 Plo 4 30/0 Us h improvd Eulr s mhod wih h = 0 o approxima h soluion o y = 4cos(x + y), y(0) = a x = 0,0,,0 Plo 5 30/ Us h improvd Eulr s mhod wih h = 0 o approxima h soluion o x = + sin(x), x(0) = 0, a = 6 30/ Us h improvd Eulr s mhod wih h = π/30 o approxima h soluion o y = siny, y(0) = 0, a = π 7 30/3 Us h improvd Eulr s mhod wih h = 0 o approxima h soluion o y = y+y 3, y(0) = 0, a = 8 3/4 By xprimning wih h improvd Eulr s mhod, find h maximum valu ovr h inrval 0, of h soluion of h IVP y = sin(x + y), y(0) = Whr dos h maximum valu occur? Giv answrs o dcimal placs Illusra wih a plo 83 Th Rung-Kua Mhod 39/7 Us h 4h-ordr Rung-Kua mhod wih h = 05 o approxima h soluion o h IVP y = y 6, y(0) = a x = Compar h approximaion wih h acual soluion valuad a x = 39/8 Sam as # for y = y, y(0) = 0 a x = 3 39/9 Sam as # for y = x + y, y(0) = a x = 4 39/0 Sam as # for y = xy, y() = a x = 5 39/ Th soluion o h IVP y = x 4 y, y() = 044 crosss h x-axis a poin in h inrval, By xprimning wih h 4h-ordr Rung-Kua mhod, drmin his poin o dcimal placs Illusra wih a graph 6 39/ By xprimning wih h 4h-ordr Rung-Kua mhod, find h maximum valu ovr h inrval, of h soluion of h IVP y = 8 x 4 y, y() = Whr dos his maximum occur? Giv your answrs o dcimal placs Illusra wih a graph 7 39/3 Th soluion o h IVP y = y x y + x + x, y(0) = 3 has a vrical asympo a som poin in h inrval 0, By xprimning wih h 4h-ordr Rung-Kua mhod, drmin his poin o dcimal placs Illusra wih a graph 8 39/4 Us h 4h-ordr Rung-Kua mhod wih h = π/6 o approxima h soluion o y = ycosx, y(0) =, a x = π 9 40/5 Us h 4h-ordr Rung-Kua mhod wih h = 0 o approxima h soluion o y = cos(5y) x, y(0) = 0, on 0,3 Illusra wih a plo 0 40/6 Us h 4h-ordr Rung-Kua mhod wih h = 0 o approxima h soluion o y = 3cos(y 5x), y(0) = 0, on 0,4 Illusra wih a plo 9 Nonlinar Diffrnial Equaions and Sabiliy S nx pag for xampls from scions

15 9 Th Phas Plan: Linar Sysms 7/ Vrify ha x() = 3, y() = is a soluion of h sysm dx/d = 3y 3, dy/d = y Plo h soluion in h phas plan 7/3 Find h criical poin for h sysm dx/d = x y, dy/d = x + y 3 7/5 Sam as # for dx/d = x xy, dy/d = 3xy y 4 7/7 Solv h rlad phas plan diffrnial quaion for h sysm dx/d = y, dy/d = x+y 5 7/9 Sam as #4 for dx/d = y x, dy/d = x + y 6 7/ Sam as #4 for dx/d = y, dy/d = x Thn plo rprsnaiv rajcoris in h phas plan 7 7/3 Sam as #6 for dx/d = (y x)(y ), dy/d = (x y)(x ) 8 7/5 Find all criical poins for h sysm dx/d = x + y + 3, dy/d = 3x y 4 Plo h dircion fild in h phas plan along wih rprsnaiv rajcoris Dscrib h sabiliy of h criical poins 9 7/7 Sam as #8 for dx/d = x + 3y, dy/d = x y 0 7/5 For h sysm dx/d = y, dy/d = x 3 x, plo h dircion ( fild in h phas plan For ach of h poins 4, ) 4, (,), (,0), conjcur whhr h soluion passing hrough h poin is priodic Plo said rajcoris 9 Auonomous Sysms and Sabiliy 57/ For h sysm dx/d = x, dy/d = y; x(0) = 4, y(0) =, skch h soluion in h phas plan and indica h dircion of moion 57/4 Sam as # for dx/d = ay, dy/d = bx, a > 0, b > 0; x(0) = a, y(0) = /6 For h sysm dx/d = + y, dy/d = 3x, do h following (a) Find and classify all criical poins (quilibrium solns) (b) Draw a dircion fild and phas porrai for h sysm (c) Dscrib h basin of aracion for ach asympoically sabl criical poin, if any 4 57/8 Sam as #3 for dx/d = ( + y)(x + y), dy/d = y( x) 5 57/0 Sam as #3 for dx/d = ( + x)(y x), dy/d = y ( + x x ) 7 57/4 Sam as #3 for dx/d = ( x)(y x), dy/d = y ( x x ) 8 57/6 Sam as #3 for dx/d = x( x y), dy/d = ( y)( + x) 9 58/8 For h sysm dx/d = y, dy/d = 8x, do h following (a) Find an quaion H (x,y) = c saisfid by h rajcoris (b) Plo svral lvl curvs of H Ths ar h rajcoris of h sysm Indica h dircion of moion on ach rajcory 0 58/0 Sam as #9 for dx/d = x + y, dy/d = x y 93 Locally Linar Sysms 57/ For h sysm dx/d = x + y + xy, dy/d = 4x y + x y, do h following (a) Vrify ha (0,0) is a criical poin (b) Show ha h sysm is locally linar (c) Discuss h yp and sabiliy of h criical poin (0,0) by xamining h corrsponding linar sysm 57/4 Sam as # for dx/d = x + y, dy/d = x + y 3 57/6 For h sysm dx/d = x x xy, dy/d = 3y xy y, do h following (a) Drmin all criical poins of h sysm (b) Find h linar sysm nar ach criical poin (c) Find h ignvalus of ach linar sysm Wha conclusions can you draw abou h nonlinar sysm? (d) Draw a phas porrai of h nonlinar sysm o confirm your conclusions, or o xnd hm in hos cass whr h linar sysm dos no provid dfini informaion abou h nonlinar sysm 4 57/8 Sam as #3 for dx/d = x x xy, dy/d = y 4 y 3 4 xy 5 57/0 Sam as #3 for dx/d = x + x + y, dy/d = y xy 6 57/ Sam as #3 for dx/d = ( + x) sin y, dy/d = x cosy 7 57/4 Sam as #3 for dx/d = xy, dy/d = x y /6 Sam as #3 for dx/d = y + x ( x y ), dy/d = x + y ( x y ) 9 58/7 Sam as #3 for dx/d = 4 y, dy/d = ( 3 + x ) (y x) 0 58/8 Sam as #3 for dx/d = ( y)(x y), dy/d = ( + x)(x y) 6 57/ Sam as #3 for dx/d = y, dy/d = x 6 x3 5 y 5