2010_09_30_M415_Q

2010_09_30_M415_Q - Name: Rec. Time: Quiz 1 Math 415.01 TA:...

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Unformatted text preview: Name: Rec. Time: Quiz 1 Math 415.01 TA: Younghwan Son _ _ Show all work; answers with insufficient support will not receive credit. GOOD LUCK! 1. (1) Draw the direction field for y' = y(4 - y) (2) Based on the direction field, determine the behavior of y as t -t 00. If this behavior depends on the initial value of y at t = 0, describe this dependency. f [y) GnsiJey ::: J (4- 'j ) -V::><7<o~ tCjl <0. 0<'1<4 if .>'>4 'J l' 'v ~~"-,> r )' 4; \ "...-, /' , ) ) ---" /'? /' / ,-/7 ~ -' ~ --;;> --;> ::> 01 ~ 'j (0) 'j LO) '7 i \.>~ 0 (0 ) ) 'je-(;)-74 tV) is JeCretJ.Jit\g. fCj} > 0 .) \.} D-IIJ if;o is if'. credJ rl\~ < 0 o.r-J o..~J ifj) .rk€Jt is Je (~OJ~ Je(l-e(J.J~ 2.Show that if a and A are positive constants, every solution of differential equation YI + ay = and b is any real number, then be-At has the property that y ----t a as t ----t 00. Hint : Consider the case a = A and a i=- A separately. ::: eo-to ejo..J6 )UtJ e B.t '1/ + lo..Je I) /1 LA. = _ e _/\t e o.t eo..c)' - J IJ 9' (at) J / -::: bJ e e (()..-N t- Cl:::).. . (e()..t 'j) I -== eat J ~ )' -= /; = ~t + c t e-o.l + ce-o-c e- 0- 70 ) At -) So 0 .J (C) -7 ewe I) b o--A y -:=.l- HeVtce j (t) 0..-f- --'10 e ()..J / -}.t I-? c)'J + ce e-)<t + c. £o..-AJ t - of o..nd e- ot ~ of • 0 0 . ...
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This note was uploaded on 07/26/2011 for the course MATH 415 taught by Professor Costin during the Spring '07 term at Ohio State.

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