M232summer08_test1_solns

# M232summer08_test1_solns - FquOX class: Problems w-‘il...

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Unformatted text preview: FquOX class: Problems w-‘il we?“ be 9+ 4144's [‘5 90 0 c9 ‘ Mae W5 I ’7 Test I Off-H 4 I MA232 Summer 2008 P W‘hm - Differential Equations [You must show work to receive full credit} 1. Classify the following differential equations: (10 pts) (a) The Mathieu equation for a vibrating pendulum: 2 935% + 27% + w3(1 + t):c=0. 2"" «Aer (lavendemr an‘dble 1‘5 x indepew den-1' \lcwx‘able 1‘5 i‘ 0355 LFV‘Ca( Variable (oev‘ﬁrrm’f homogeneou5 (b) The Fisher—Kolmogorov Equation (found in combustion and crystallization problems) 2 Qa—:=u(1 — u) + 3—53. 2'“, orécf aerJen-l VOWNble ,‘5 LL Mega/«Jan Van‘mbl€5 are, 31 and f PDE Non lmemr nanlfnear fevm f5 UL.(I"‘*) 2, Verify that y is (or is not) a solution to the differential equation. (note: only 1 of the 3 is not) (10 pts) (a) t+yy’=0, t2+y2:1. \IZ: l‘tz 79W 1- t2 -l 7' =1; ~2t) (Pt?) /Z ‘12 ’I’ i + (m?) (thi’ it) V45 2 "0 (b) y”'=et, y=et+at2+bt+a F \g'> 65+ Zat + L \ ‘I "3 6’5" Z a > jut: at 1/15 m (c) y’+y=t2—2, y=Cet+t2—2t. \VE cet+zt»z I} 15 Ce &ﬂ~z+ (c 4 52%: = 52,2 Xho 3. Verify that y is a solution to the differs the initial condition. (10 pts) ntial equation and determine the value of C which satisﬁes (a) t3 + yew/=0, t“ + 314:0 (y > 0), y(0)=1- a. bad: m +0 Check 7Q( + 0‘" ‘/ v= (c— t“) “ (b) ty’=3y,[\ y=0t3, y(—4)=16 KKK: 5(£2 z 3 Wow I~C— /_ 1‘ Ci = at 3 a "L 3 )/ ‘62::(44) —7 6’ H . S m \\\\\\\\\:////II/II: . 1|] 4 ‘ s % \\\\\\\\\._///IIIIII , \\\\\\\\\._////IIIII e w _ I. r ‘tittziiiié “MHZ”: ZZZ”, r. .\\\\\\\\\..///IIIIII VIII/Ilz’.— ’xllllll u \\\\\\\\\.—////I//II42 III/III’._ a 2 W \\\\\\\\\://///IIII III/11:: ’ Z: \\\\\\\\\_r///IIIIII I —lIIII///—_ I 2 .__— .mw \\\\\\\\\—_////IIIII ltlIII//’_ / _—.. a \\\\\\\\\..///IIIIII .lllIlIl/a_ I . ..__ tam \\\\\\\\\_./xlllllll llllllllz_ I '——— d \\\\\\\\\..////IIIII AIIIIIIIIII, I a a’f’ \\\\\\\\\.—///IIIIII 4 V. lllllll‘\~ .\ 1”; w \\\\\\\\\~—/IIIIIIII II.||.|I.\\\— \ 1/ I S|I\.\\\\\ 4 I d \\\\\\\\\——////IIIII 2 . \ . d \\\\\\\\~..z//IIIIII. “ \\\ \ l on \| \\\ \x..///IIIIII._... \\\\\\\~._ ~ 4 \\\\\\\~'r///IIIII |\\\\\\~.. \ m ﬁ\\\\\\\\\“.///llllll \\\\\\\-_ x d . \\\\\\\\\ .IIIIIIIII4 \\ \\\\~.. ~ 1 \ t rl Til. . C » a .. . a 4 4.. «Iii \\\\\\\~.. ~ ._ 6 pm A u a .. p 0 L4 ¢ 6 t c B e r. r. O C e h t h t .m n m s I I’ h 9 . 0 Lu Jl h 0 2 _1 7? C, “HM: arrec‘n‘an I / y :“y 4. Match the differential equat (10 pts) y/ / 3/ =9; 8) M 6N‘ec‘h‘on grab 6(Wés OW i éfpewdS J w PT n. d F ‘T c e C V IO M A 369mb on ‘5 WM 5. Solve the initial value problems by separation of variables. (15 pts each) (a) ty’ — 11:0, y(2)=2. 6 ~ 0‘ Ir C - f at ’ ‘l [K 2>cm ~> 0’ \ ‘ , ______ 3‘4""‘ t” (b) gal—ﬂ, y(1)=—2. Pcoblem ti )7 Wk homewoFK 437/ on “WK pay 6 \ PM 0% class: Probkems \ aca ,‘n \[adr nai‘es. (c) y'=ty3(1 + W1”, y(0)=1. “(z ,_\_ ‘ 1 v %ctﬁg(l*tz) 117- ‘ W‘Z“ ’ l 2-: .— ‘ \I 1 [0“ t‘)'z-§ cl ’ \ ‘ gﬁhkty M: _,/Z 117’ “Ft-“Ll 1;:%_Z[0+t23/2+C13 ‘9": T; :6,“ a) I- . t 1‘6“? 31 (p1 C 4/1 I: Ii’Z(H()B z'zam) a (,2 ‘72 "° iéﬂiww nm ® 0(6? pd+ 7A 3-(‘ J16; ‘3 fwuemulq m “~ng ‘ answe( (d) 312%? 9(2)=0 531 u T U“ ‘ ’— “ Z (375 I111 Kt: ‘ [Winn-‘1) ‘14”1 z I» ‘f _ sfv’li JHL‘RQ‘A 611/2 0: '*W) i 2— Z he, CD WONG ...
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## This note was uploaded on 08/31/2009 for the course MA 232 taught by Professor Toland during the Spring '08 term at Clarkson University .

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M232summer08_test1_solns - FquOX class: Problems w-‘il...

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