Mah 14 Soluion Te 1 B Spring 016 Problem 1: Ue eparaion of ariable o ole he Iniial alue DE Soluion (14p) e =, (0) = 0 d = e e d e d = o = ln e d uing u-du b leing u = e 1 e = + where C = for he iniial alue. e + 1 e =, (0) = 0 Problem : Gien ha = + 3 i a unique oluion for he iniial alue DE 3 e + = =. Find p(), g() and. 0 p ( ) g ( ) and (0) 0 Soluion (10p) conider he oluion : = e + 3 fir we hae = e + 0 3 = 1 3 = 3 3(0) 0 3 Now diide b e o ge ha 3 3 3 e = 1+ e 3 e now conider he deriaie of boh ide e = + e + e e = e + = 3 3 3 3 3 3 [ ] 0 3 9 [3 1 9] e [3 8] From he proce of oling he fir order linear DE we know ha he 3 3d inegraing facor i u = e o p() = 3 and g() = 3 8 = e 1
Problem 3: Gien he iniial alue problem ealuae (-) when ep alue i h = 1. = 0, ( 4) =, ue Euler Mehod o Soluion (10p) = 0, ( 4) = uing he poin (-4, ) and he lope m = 1 = he equaion of he angen line i = (+) and ( 3) = 1 Nex uing he new poin ( 3,1) and lope m = 3 = he equaion of he angen line i = 3 8 and ( ) = ec Problem 4: Wha i he large -ineral for which ( 4) = ha a guaraneed + unique oluion for iniial alue (-1) = 3 according o he Exience Uniquene Theorem? Soluion (8p) ec ( 4) = + ec = ( 4)( + ) ( + ) 4, 0,, ( nπ ) and (-1) = 3 The large ineral for hi iniial alue i (- π, 0)
Problem 5: An objec weighing 19.6 Newon i dropped downward wih an iniial eloci of 0mec. The magniude of he force on he objec due o air reiance i R = S0. ( S i peed.) a) I reiance a poiie or negaie force in hi cenario? b) I eloci a poiie or negaie force in hi cenario? b) Se up he differenial equaion ha model hi cenario in erm of peed. (Do no ole) c) Se up he differenial equaion again in erm of eloci and gie iniial alue. (Do no ole) Soluion (1p) a) Reiance i poiie b) Veloci i negaie c m mg ) = + 0 = + 19.6 0 = 9.8 + 0 = = 0 = 9.8 40 d) 19.6 ince = ince eloci i negaie Problem 6: Conider he iniial alue non-linear DE = wih (0) = 0 ( 1) I a unique oluion guaraneed for hi iniial alue? Juif our concluion. = wih (0) = 0 ( 1) In order o deermine he open region ha would guaranee a unique oluion for hi iniial alue we will deermine where he following i coninuou. 1) ) f(, ) = = where 1, 1, 0 ( 1) f 1 = where 1, 1, 0, > 0 ( 1) Howeer he parial deriaie i no defined for he iniial alue o he oluion i no guaraneed. 3
Problem 7: Below i an example of he logiic equaion which decribe growh wih a naural populaion ceiling: Deermine he differenial equaion below ha i repreened b hi direcion field. Explain our reaoning for our choice. Explain wh ou rejeced he oher poibiliie. a) ( 15) = b) 15 ( ) = c) neiher a) or b) The equilibrium oluion occur a = 0 and = 15 for boh a) and b). Now e oher iocline: For a) le = 0 o ge lope = 0(0 15) = 100 which i poiie bu he acual lope i negaie. Now e b) a = 0 o ge = -100 which i he correc ign. Now we need o check b) a oher iocline o be ure ha i reall i he correc DE. Te = 10 o ge = 50 Which i poiie and mache. Now e = -10 o ge = -50 and ha he correc ign. Therefore b) i he correc. 4
Problem 8: Sole he following fir order linear DE uing he inegraing facor. + = + 1, (1) = 1 Soluion (14 p) + = + 1, (1) = 1 + 1 and inegraing facor i u() = e e e + 1 o [ ] d = d = + + C 1 1 5 1 1 5 = + + C uing iniial alue C = and = + ( ) d ln( ) ln( ) + = = = = Problem 9: A ank ha a capaci for 800 gallon and conain 300 gallon of waer wih 7 lb of al iniiall. A oluion conaining of 6 lbgal of al i pumped ino he ank a 10 galmin. A well irred mixure flow ou of he ank a he ame rae. a) Se up he differenial equaion ha model hi iuaion. b) Sole b uing he inegraing facor o find he equaion for he amoun of al in he ank a an ime. (how all work in deail) c) Find and explain he limiing alue of our oluion. d) Suppoe he rae a which he oluion flow ou change o 8 galmin, Se up he differenial equaion ha model hi new iuaion. (Do no ole hi DE.) e) Deermine how long i will ake o fill he ank uing he iuaion in par d). 5