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# J.J._Salacuse_Exam_1_Answer_Key_Sprint_2007 - = 0 has...

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MATH-204, Exam] May 4, 2007 Dr. J. J. Salacuse Name(pRlN1) ..If"us k"V # 1. (a)Give the region in the x-y plane for which the following IVP has a unique solution passing through the point (xo' Yo)' JustifY vour answer. Note you do not have to solve the de. (8 points) , /'101; ,-e. x'S'~PI~. UtlJ' ]<Je .... vt~.s r I~ 11,(1 g- (C) l> 1 ""') -titu t'4;> (!!~ ~. MP~. -ti .""1 ~_. tt.90/. ~-:c" to. C-\r\o+~~~ol '\$; J:v t' ~ ("X) =0 @ ...0:> 3 (j ~ = ?( +- <. 'I -) (? l~:i X-K:. ,v\, i+'~~. Co 1"\ &'l!-' (),.., -:::-~ 001 O;-;Z:-v P (1'o<=;j >; ~ ~) ~~ ByJ 1

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#2. Derive the general solution (explicit form) of the logistic equation d P = a pet) - b p2(t) , (20 points) dl I -~ /= J1(:.t-~ f) t-.J§ P e)~ PJ 4 --=~ 1/::::- Y<:< ?-= ~ I~ JS ~ -==) t ~~ ~ 2
~& = (b) Obatin the general solution to the hmgn. de d y - 2 x y = O. (8 points) dx '"l .... 7< (j = C.e #3.(a) Obtain the general solution to the de. d y - 2 x y = x (12 points) dx -l- L ~ -1. Z X)dfl( -.J( L ~I T~c {)D,- ~ = ~ ?.. _((. "",1.. ~ , -,.. K e /j-e 2Xr;~~ X .•.• ,-y2- J - ')( ~ - ", '*11 r~ :7' ) :: C )( e-~ -= J ~?r~ ~X _/C '2-+-C -2- -t~+-CC"('- ---- @ 3

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#4. The de x 2 d2 y + 2 x dy - 6y = 0 has sOlutiO{-;~
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Unformatted text preview: = 0 has sOlutiO{-;~ X'~~d the other l.i. solution.(20 points) d x 2 dx '-'--. .~--. II b ~) ('J + fr (J-X"2.-;;?-= (j 2--:0 l)(~)(, 'L I <41 oJ ~ • ft-, • I•••• c:s: ~ R. (}" oJ e...r U ., [,XJ-+ LJ f [2. 2. )(-+ ~ ?( 1.J =- ~ )( :::--., 11 r 1..] '[ . =..; U )( f U tI'K]-:. ~ Ul/[X]-r l/'ra.l-=c". @ u ,-= ;>(-4 . f'l-b-&~~-RM)(-e. .. ~)J2 =-,3 ... { 01 (tx-1 J/c.(J C-cL0 4--7 -.__ I.- \ ~(bJ (b) Solve the IVP consisting of the de in part(a) and the initial conditions y(O) = 1 and dY5:/ = 2 (6 points) V (<i) r-C I ". I~ ~,-::;:f\ [e, (oS (,V3y. .).,. c,,,-d' ~~ 'Y:]e 2(_2) ~X I,-'Yr r::" I.' ,+ e... J:-L, "3SfV\ eVXx) r C2,. . 6--c~s (Jj')')] ":-C, (-Z) + Cz. .-.B-= L-~"C:2§-== 4~v' 5~~ 'o/w!.T~ (c) Show a rough plot of the solution in the vicinity of the initial point. (4 points) 5...
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