apm255fa09exam1solutions

apm255fa09exam1solutions - APMZSS Sept. 29, 2009 @4091 Exam...

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Unformatted text preview: APMZSS Sept. 29, 2009 @4091 Exam 1 Instructor: w. Zhang Instructions: Close books and notes. Show your work and justifications. Calculator for performing computation, integration table, and blank scratch paper are permitted. Student’s Name: Mark: 1. (12 pts.) Identify if the following ordinary differential equations (ODEs) to be (i) linear or noulinear, (ii) separable or non-separable, and (iii) autonomous or non- autonomous. . ‘ A‘gflxz _—.-. 2 41—2 ____ FL fZ/QWM 3'; it; xiii—«2o; Ebb/W7! WWW .—__ 1 (H?) P®)w’—(y-l)2=o :37 24/; 517 z. a- —— as 9' (1/ 2 ‘5‘7’3/7fi/M/ééa {[2}? M/WMW5 2. (40 pts.) Solve THE general solution (i.e. all solutions) of the following ODEs. (a)xy'—2y+x‘=0 7? flbécégzaz‘i 63f t (b)y'—2y(4—y)=0,__ fill/h: ( __ iii? 2 J #7 fl 973 ‘7’?) 27f g + fé—jy/g :M‘Z‘C .-~ "$7 :5”: JMW"2§C¢W}:QZ+C :7 if” :‘QZ’IL‘C- fl/aflz’rw 2:220 —-=> 5’26 51%. j/fiflfiflm ?’ /:o?,Z’+C, 3. (18 pts.) Find the particular solution of the initial Value problem y’-2Iy2*3x2y2 =0, y(1)=0. fl/ 2- fl’:@z r32“)? ‘2’ :37 27% zéz 2-52”) 2 :fézz—t-jzjflf—g, .27 Fifi-:1 +2 +C, 4. (15 pts.) Find all equilibrium solutions (ie. all critical points) of y' = y( y + 2)( y —3)2 and determine whether each equilibrium solution (critical point) is stable, unstable or semistable. &:g/=2(3+2)(g*3)2' :pg/aeg £223] £23 M X/ ,‘ (73475252! . r— H ““ M F‘— /7/% 797/5/‘4/1’1' f" / 5 igyii) 5‘ 0 0 a We) 2(4) main “2/ p 3 9; 3%) :«w (0270 5. (1 5 pts.) Suppose that at time 2' = 0 one person in a class of 43 people has heard a certain rumor. After 1 week the number 13(1) of those in the class who have heard the rumor has increased to PG) = 3. Assuming that P(r) satisfies a lo 'stic e nation when will exactly 22 people in the class have heard the rumor? When will all people in the class have heard the rumor? flag/I Meéa/ flag—3 3;;— s/em *7?) #275677) {fig/g %/m’f :7 wig/Frflfc #flFF-VZ’I/fihf-Fflgfl—X-C #fiflfi wflcrgzc jam {7(0):}. 2%: 76%; *—7 $5425 'L/flfijzk +é/fle) .fl. 7...}... . ’77 K 43%” 3'35?” =§j¢flg 25.99, agééS” - %/é’// flaw? jug -—.,/»7.24= Z: flea: 7.; #2. fig- 7&4?» M H fig; “ Jig? : o/zw,éi fizsz/yggé) %%47 f”? “3/ ...
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apm255fa09exam1solutions - APMZSS Sept. 29, 2009 @4091 Exam...

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