MA Study Guide #1
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1 MA 66 Su Guide #1 (1) Special Tpes of Firs Order Equaions I. Firs Order Linear Equaion (FOL): + p() = g() Soluion : = 1 µ() [ ] µ()g() + C, where µ() = e p() II. Separable Equaion (SEP): dx = h(x) g() Soluion : 1 g() = h(x) dx (The soluion is usuall given implicil b he above formula. You ma ge addiional soluions from g() =. You mus check o see if here are exra soluions.) III. Homogeneous Equaion (HOM): dx = f(x, ), where f(x, ) = f(x, ) Soluion : Le v = x Subsiue hese ino dx. Hence = xv and dx = xdv dx + v. = f(x, ) o obain a Separable Equaion. IV. Exac Equaion (EXE): M(x, ) dx + N(x, ) =, where M = N x Soluion : Soluion = ϕ(x) given implicil b ψ(x, ) = C where : ψ x = M(x, ) = ψ = M(x, ) dx + h() ψ ψ = N(x, ) = = ( ) M(x, ) dx + h() = f(, ) has slope f(, ) a he poin (, ). The direcion field (or slope field) of he d.e. indicaes he slope of soluions a () Direcion Fields. A soluion = ϕ() o he d.e.
2 various poins (, ). The direcion field ma be used o give qualiaive informaion abou he behavior of soluions as (or, or, ec). Direcion fields ma also be used o approximae he inerval where a soluion hrough a poin (, ) is defined. (3) Applicaions of 1s Order Equaions. (A1) Mixing Problems : Q() = amoun of subsance in soluion a ime dq = Rae In Rae Ou = r ic i r o c o (A) Exponenial Growh/Deca : Q() = quani presen a ime dq = r Q (A3) Newon s Law of Cooling : T () = emperaure a ime dt = k (T T a) (T a = ambien emperaure) (A4) Falling & Rising Objecs : You should be able o se up and solve simple problems using Newon s nd Law: F = m dv. Near he surface of he Earh, he force due o gravi is he weigh of he objec F g = mg. Le F d = magniude of drag force. (a) For falling objecs, we usuall le he posiive direcion be he downward direcion so m dv = mg F d. (b) For rising objecs, le he posiive direcion be upward. For he upward porion of he fligh, m dv = mg F d ; while for he downward porion of he fligh, m dv = mg + F d. (a) Falling Bo (b) Rising Bo F d m F = mg g + direcion F = mg g m F d (i) rising par + direcion F = mg g F d m (ii) falling par
3 (4) Exisence and Uniqueness Theorems for 1 s Order Equaions. (a) THEOREM (Firs Order Linear). If p() and g() are coninuous on an inerval { α < < β conaining, hen he IVP + p() = g() has a unique soluion ( ) = = ϕ() on he inerval α < < β, for an. Noe: The larges such open inerval conaining is where he soluion = ϕ() is guaraneed o exis. (b) THEOREM (Firs Order Nonlinear). If f(, ) and f are coninuous in some recangle R: { α < < β, and γ < < δ and (, ) lies inside he recangle R, hen he IVP = f(, ) has a unique soluion on he inerval ( ) = h < < + h, for some number h >. Noe: The number h is no eas o find. The inerval conaining where soluion exiss can be esimaed b looking a he direcion field of he differenial equaion. To deermine he exac inerval, ou mus solve he IVP explicil for. (5) Auonomous Equaions: Equaions of he form = F () ( ) are said o be auonomous since does no depend on he independen variable. Such equaions can have consan soluions (i.e., = K) which are called equilibrium soluions. These soluions are found b solving F () =. (These are also called criical poins.) You should be able o find all equilibrium soluions o he auonomous d.e. ( ) and skech nonequilibrium soluions using he phase line of he differenial equaion ( ). You should also be able o classif he sabili of he equilibrium soluions as follows: (a) Asmpoicall Sable - Soluions which sar near = K will alwas approach = K as : = K (b) Asmpoicall Unsable - Soluions which sar near = K does no alwas approach = K as : = K
4 (c) Semisable - This is a special pe of unsable soluion. In his case soluions on one side of = K will approach = K as, while soluions on he oher side of = K will approach somehing else: = K Remark. To skech non-equilibrium soluions of ( ), ou do no necessaril need direcion fields, ou can use ordinar calculus. Since = F (), he graph of F () vs will deermine where he soluion = ϕ() is increasing (F () > ) or decreasing (F () < ). B he Chain d df () Rule, = F (), hence a graph of df F will deermine where he soluion = ϕ() is concave up (F F > ) or concave down (F F < ). = f(, ) (6) Euler (Tangen Line) Mehod. Approximae acual soluion ϕ() o ( ) = using he Euler (Tangen Line) Mehod : n = n 1 + h f( n 1, n 1 ) where h = sep size. A each ieraion, k ϕ( k ), where k = + hk. (7) Second Order Linear Homogeneous wih Equaions Consan Coefficiens. The differenial equaion a + b + c = has Characerisic Equaion ar + br + c =. Call he roos r 1 and r. The general soluion o a + b + c = is as follows: (a) If r 1, r are real and disinc = C 1 e r 1 + C e r (b) If r 1 = λ + iµ (hence r = λ iµ) = C 1 e λ cos µ + C e λ sin µ (c) If r 1 = r (repeaed roos) = C 1 e r 1 + C e r 1 (8) Theor of nd Linear Order Equaions. The Wronskian is defined as 1 () W ( 1, )() = 1() () () (a) The funcions 1 () and () are linearl independen over a < < b if W ( 1, ) for a leas one poin in he inerval. (b) THEOREM (Exisence & Uniqueness) If p(), q() and g() are coninuous in an + p() + q() = g() open inerval a < < b conaining, hen he IVP ( ) = ( ) = 1 has a unique soluion = ϕ() defined in he open inerval a < < b..
5 (c) Superposiion Principle If 1 () and () are soluions o he nd order linear homogeneous equaion P () + Q() + R() = over he inerval a < < b, hen = C 1 1 () + C () is also a soluion for an consans C 1 and C. (d) THEOREM (Homogeneous) If 1 () and () are soluions o he linear homogeneous equaion P () + Q() + R() = in some inerval I and W ( 1, ) for some 1 in I, hen he general soluion is c () = C 1 1 () + C (). This is usuall called he complemenar soluion and we sa ha 1 (), () form a Fundamenal Se of Soluions (FSS) o he differenial equaion. (e) THEOREM (Nonhomogeneous) The general soluion o he nonhomogeneous equaion P () + Q() + R() = G() is () = c () + p (), where c () = C 1 1 () + C () is he general soluion o he corresponding homogeneous equaion P () +Q() +R() = and p () is a paricular soluion o he nonhomogeneous equaion P () + Q() + R() = G(). (f) Useful Remark : If p1 () is a paricular soluion of P () + Q() + R() = G 1 () and if p () is a paricular soluion of P () + Q() + R() = G (), hen p () = p1 () + p () is a paricular soluion of P () + Q() + R() = [G 1 () + G ()]. Pracice Problems 1. Deermine he order of each of hese differenial equaions; also sae wheher he equaion is linear or nonlinear: (a) + x = 1 (b) x + = 1 (c) ( ) 3 + = 1 (d) + = 1. (a) Which of he funcions 1 () = and () = is/are soluions of he IVP =, () =? (b) Which of he funcions 1 () = and () = are is/soluions of he IVP =, (1) = 1? 3. For wha value(s) of r is = e rx a soluion of =? 4. (a) Show ha = x 3 is a soluion of he iniial value problem = 3 /3, () =. (b) Find a differen soluion of he iniial value problem. 5. Find an explici soluion of he iniial value problem x =, (1) = 1. Indicae he inerval in which he soluion is valid. 6. (a) Find an implici soluion of he iniial value problem = x, () =. + 1 (b) Find an explici soluion of he iniial value problem = x, () = For wha value(s) of a is he soluion of he IVP + e =, () = a bounded on he inerval? 8. Deermine wheher each of he following differenial equaions is linear, separable, homogeneous, and/or exac or none of hese. (a) x + + (x + 3) dx = (b) x (x + ) dx =
6 (c) (x )dx + (x + + 1) = (e) ( + 1) + (x + 1)dx = (d) x (x + 1) dx = 9. Find implici soluions o (a) x + x = (b) (1 + ) dx x = (c) x + + x = 1. Find an implici form of he general soluion of he differenial equaion dx = x +. x 11. Find an implici soluion of he IVP x (x + ) =, (1) = 1. dx 1. If x + (x + 1) = xe x and (1) =, hen () =? 13. Use he given direcion field o skech he soluion of he corresponding iniial value problem = f(, ), ( ) = for he indicaed iniial value (, ) : (a) (, ) (b) (, ) (c) ( 1, 3) (d) (, 4) 14. For each of he iniial value problems deermine he larges inerval for which a unique soluion is guaraneed : (a) = 1, (1) = (b) + (an ) = sec, () = (c) + x x 9 = 1 x, () = 1 (d) (x + 4) x = 1 x, ( ) = For each of he iniial value problems deermine all iniial poins (, ) for which a unique soluion is guaraneed in some inerval h < < + h: (a) = +, ( ) = (b) = /, ( ) = (c) = +, ( ) = 1 (d) = 1/3 + 1/3, ( ) = (e) =, ( ) = 16. Find he explici soluion of he iniial value problem = 1, () =. Where is his soluion defined?
7 17. Suppose is proporional o, () = 4, and () =. Se up and solve an iniial value problem ha gives in erms of. For wha value of does () = 3? 18. A hermomeer reads 36 when i is moved ino a 7 room. Five minues laer he hermomeer reads 5. Se up and solve an iniial value problem ha gives he hermomeer reading minues afer i is moved ino he room. Wha will i read en minues afer i is moved ino he room? 19. A ime = a 5 gallon ank conains 4 pounds of sal mixed in 1 gallons of waer. A soluion ha conains 3 lb of sal per gallon of soluion is hen pumped ino he ank a a consan rae of 5 gal/min. The well-sirred mixure flows ou of he ank a he rae of 3 gal/min. Se up and solve an iniial value problem ha gives he amoun of sal in he ank afer minues. Wha is he concenraion of sal in he ank a he ime he ank becomes full?. A huge 3 gallon radiaor is full of a 6% anifreeze soluion. Pure waer is poured in a a rae of 5 gal/min and he sirred mixure is drained a he same rae. How long do we pour waer ino he radiaor o ge a 5% anifreeze soluion? 1. Se up and solve an iniial value problem ha gives he verical veloci of a 18-lb parachuis seconds afer she jumps from an airplane ha is fling horizonall a an aliude of 5 fee. Assume ha air resisance is eigh imes he speed and ignore horizonal moion and downward direcion is posiive.. Consider he differenial equaion (a) Wha are he equilibrium soluions? = ( ). (b) Which equilibrium soluions are sable/unsable? (c) Skech he graph of he soluion of he differenial equaion for wih each of he iniial values () = /3, () =, () = /3, () = 4/3, () =, () = 8/3. (d) Find he explici soluion of he iniial value problem = ( ), () =. (e) For wha values of is he soluion in (d) valid? 3. Consider he differenial equaion = F (), w w = F ( ) where he graph of F () is indicaed below. (a) Wha are he equlibrium soluions? (b) Which equilibrium soluions are sable? (c) Skech some soluions o = F (). 4. Esimae he soluion a = 1.5 o he IVP = 5, (1) = using he Euler Mehod wih h =.5. Wha is he rue soluion a = 1.5? 5. Find he general soluion o (a) = (b) =. = 6. For wha value of α will he soluion o he IVP () = α saisf as? () = 7. Find he larges open inerval guaraneed b he Exisence and Uniqueness Theorem for which he iniial value problem 3x x = 1 x 3, (1) = 3, (1) =, has a unique soluion.
8 Answers (1) (a) 1 s order nonlinear (b) 1 s order linear (c) 1 s order nonlinear (d) 3 rd order linear () (a) 1 and (b) 1 onl (3) r =, r = 3 (4) (a) = 3x = 3(x 3 ) /3 = 3 /3 ; = 3() /3 (b) (5) = x x + 1, x > 1 (6) (a) + = x (b) = 1 + 4x + 1 HOME (7) a = 1 (8) (a) HOME and EXE (b) (c) none of hese pes (d) FOL and EXE (e) SEP and EXE (9) (a) (HOME) ln 1 ( x ) = ln x + C and = x and = x (b) (SEP) + 1 = Cx (c) (EXE) x + x = C (1) 1 ( ) = ln x + C (11) x + x + = 1 x (1) () = 3 e (13) See below : (14) (a) > (b) π < < π (c) 3 < x < (d) 4 < x < (15) (a) all (, ) (b) all (, ) wih (c) all (, ) (, ) (d) all (, ) wih (e) all (, ) where 1 < < 1 and (16) = 1 + 3ex 1 3e, soluion defined for 1 ln 3 < x <. x (17) (18) (19) = k () = 4 () = T = k(t 7) T () = 36 T (5) = 5 Q = 15 Q() = 4 ; = 4e (ln.5)/, = 3Q 1 + ln.75 ln.5.83 ; T = 7 34e (ln(1/17))/5, T (1) 58. ; Q = 3(1+) 6, (1+) 3/, Q() 5.95 lbs/gal
9 () If Q() = # gals of anifreeze, hen Q = Q, Q() = 18 and so Q() = 18e 6. newline 6 Hence = 6 ln minues 6 4 dv = 18 8v (1) ; v = 16(1 e ) v() = () (a) = and = (b) = is sable, = is unsable (c) See below (d) = (e) The soluion is valid for all if ( )e. If > or <, he soluion is valid onl for < < 1 ( ) ln. (3) (a) = 1 and = 3 (b) onl = 1 is sable (c) See below (4) =.375, rue soluion ϕ(1.5) = 1 5 (13 58 e.5 ).396 (c) 1 3 (c) (5) (a) = C 1 e + C e (b) = C 1 e cos + C e sin (6) α = (7) < x <
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