Math 2214 Solution Test 1A Spring 2016

Transcription

1 Mah 14 Soluion Tes 1A Spring 016 sec Problem 1: Wha is he larges -inerval for which ( 4) = has a guaraneed + unique soluion for iniial value (-1) = 3 according o he Exisence Uniqueness Theorem? Soluion (8ps) sec ( 4) = + sec = ( 4)( + ) ( + ) 4, 0,, ( nπ ) and (-1) = 3 The larges inerval for his iniial value is (- π, 0) Problem : Given he iniial value problem evaluae (-) when sep value is h = 1. = 0, ( 4) =, use Euler s Mehod o Soluion (10ps) = 0, ( 4) = = using he poin (-4, ) and he slope m = 1 he equaion of he angen line is = (+) and ( 3) = 1 = Nex using he new poin ( 3,1) and slope m = 3 he equaion of he angen line is = 3 8 and ( ) = 1

2 Problem 3: Solve he following firs order linear DE using he inegraing facor. + = + 1, (1) = 1 Soluion (14 ps) + = + 1, (1) = and inegraing facor is u() = e e e + 1 so [ ] d = d = + + C = + + C using iniial value C = and = + ( ) d ln( ) ln( ) + = = = = Problem 4: Given ha = + 3 is a unique soluion for he iniial value DE 3 e + = =. Find p(), g() and. 0 p ( ) g ( ) and (0) 0 Soluion (10ps) consider he soluion : = e + 3 firs we have = e = 1 3 = 3 3(0) 0 3 Now divide b e o ge ha e = 1+ e 3 e now consider he derivaive of boh sides e = + e + e e = e + = [ ] [3 1 9] e [3 8] From he process of solving he firs order linear DE we know ha he 3 3d inegraing facor is u = e so p() = 3 and g() = 3 8 = e

3 Problem 5: Use separaion of variables o solve he Iniial value DE Soluion (14ps) e =, (0) = 0 d = e e d e d = so = ln e d using u-du b leing u = e 1 e = + where C = for he iniial value. e + 1 e =, (0) = 0 Problem 6: Consider he iniial value non-linear DE = wih (0) = 0 ( 1) Is a unique soluion guaraneed for his iniial value? Jusif our conclusion. = wih (0) = 0 ( 1) In order o deermine he open region ha would guaranee a unique soluion for his iniial value we will deermine where he following is coninuous. 1) ) f(, ) = = where 1, 1, 0 ( 1) f 1 = where 1, 1, 0, > 0 ( 1) However he parial derivaive is no defined for he iniial value so he soluion is no guaraneed. 3

4 Problem 7: Below is an example of he logisic equaion which describes growh wih a naural populaion ceiling: Deermine he differenial equaion below ha is represened b his direcion field. Explain our reasoning for our choice. Explain wh ou rejeced he oher possibiliies. a) ( 15) = b) 15 ( ) = c) neiher a) or b) The equilibrium soluions occur a = 0 and = 15 for boh a) and b). Now es oher isoclines: For a) le = 0 o ge slopes = 0(0 15) = 100 which is posiive bu he acual slope is negaive. Now es b) a = 0 o ge = -100 which is he correc sign. Now we need o check b) a oher isoclines o be sure ha i reall is he correc DE. Tes = 10 o ge = 50 Which is posiive and maches. Now es = -10 o ge = -50 and has he correc sign. Therefore b) is he correc. 4

5 Problem 8: A ank has a capaci for 800 gallons and conains 0 gallons of waer wih 7 lbs of sal iniiall. A soluion conaining of 6 lbsgal of sal is pumped ino he ank a 10 galsmin. A well sirred mixure flows ou of he ank a he same rae. a) Se up he differenial equaion ha models his siuaion. b) Solve b using he inegraing facor o find he equaion for he amoun of sal in he ank a an ime. (show all work in deail) c) Find and explain he limiing value of our soluion. d) Suppose he rae a which he soluion flows ou changes o 8 galsmin, Se up he differenial equaion ha models his new siuaion. (Do no solve his DE.) e) Deermine how long i will ake o fill he ank using he siuaion in par d). Soluion (16 ps) Q a) Q = 60 for Q(0) = 7 d Q Q + = 60 where u = e = e so Qe d = 60e d b) Qe = 1800e + C 1800 Ce hen using he iniial value C = 1793 Q = + so Q = e c) Limi Q = 1800 lbs of sal d) Q(8) Q = 60 for Q(0) = e) Volume = 0 + se 800 = 0 + and solve for o ge = 50 min. 5

6 Problem 9: An objec weighing 19.6 Newons is dropped downward wih an iniial veloci of 0msec. The magniude of he force on he objec due o air resisance is R = S0. ( S is speed.) a) Is resisance a posiive or negaive force in his scenario? b) Is veloci a posiive or negaive force in his scenario? b) Se up he differenial equaion ha models his scenario in erms of speed. (Do no solve) c) Se up he differenial equaion again in erms of veloci and give iniial value. (Do no solve) Soluion (1ps) a) Resisance is posiive b) Veloci is negaive ) s = + 0 s c mv mg v = v s = v v = v = v 0 v v = d) 19.6 since s = since veloci is negaive 6