240-2006&2007-3-F10-August2007

240-2006&2007-3-F10-August2007 - Kuwait University...

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Unformatted text preview: Kuwait University, Dept. OfMatli & Comp. Sci. Math 240: Ordinary Differential Equations Sunday, August 5, 2007 Final Examination Duration : Two Hours Answer all questions. Cell phones and Calculators are NOTallowed. Max. Marks:40 [ Each questions 5 marks ] 1. Obtain the general solution ofthe differential equation: )7 — (Bx—y + ex = 0 . . . 2 —- 2 2. Find the general solution of x (x _1)y — y — xlx -— 2)}; = 0 . 3. Find the particular solution of x v + 1 — 2x2 ' — 4x 3 4x y y y a 2 given y, 2 eJr is a solution ofthe corresponding homogenous equation. 4. Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions: ty" (ti—Zy'lrl-tyltlws yl0)=0;y’(0)=1- 5. Solve the following equation for y = y (i), using convolution theorem for the Laplace transform, 1 MI) =l 0-“) W) du y(0)=0:y'l0)=l‘ 0 6. Find the power series solution ofthe differential equation valid near origin and calculate thefirst Five nonzero terms: (x2 +4)y"+xy'~9y= 0. 7. Evaluate (a) L" (s e“): i b L "y in cost— d (s+1)sz+25+2 (J [Jye s y ( y) y] 3, 05t<7t/2 in terms of the 0: function and find L[F 0)]. 005 t , t 2 iT/Z 8. (i) Express F(t)=|: (ii) Showthat L [if F(u)du] dy [email protected] 2 0 n 3 Good Luck OO 99 DIIFFE-RENTIAL EQUATIONS (MATH 240) INVERSE OPERATOR FORMULA 1 a WRWJFWJ EXPONENTIAL SHIFT iJwvaEHD—mwm ____J LAPLACE TRANSFORM FORMULA 1- LVN} = fiS) 2. L{F’(t)} = 5m) _ F(0) 3. L{F”(t]} = 5mg) m 5H0) H No) k:n——l kzo a“? (S) = L{(-t}”F(f-)} LUWWH=fiflfli zsmhwmm _i INVERSE LAPLACE TRANSFORM L4Um—m}=me |_. 4 G" I'— Jml r2 li”{e‘“f[s)} = 04(t —- C)F(t — C) 3 '7 5 6 - L4wam}=ffimGU—mw L— n 13. L‘1{52—:-k—2} : cosh(kt) Ly.menhdfflmw OO 99 i""\<7t-% - 324/29 FE CCCCCC-C Cm C i} .C. (£7 " .797) ”K 4- y a CC: C: __ «w ; C1 C21- 5 C C) C 4‘ . 9,1,; . _ 2" “LC - e “ (’JL 1 §%:T;' L" (”)4 "i C t (k I {"3" (9'37” 1 . h 5"“ I ,3 MC C— N) ., Cl ‘ ~~~~ 52-» "XZ'C 2L 4“ )y - VVVVV ’— "2C Cry—2)) 3-3 0 Lu C , r , C #3 J a}? my} fl " V1 ’8:th 3- )‘I’j J1 , 1/ ~ “'1"; H (.1 : CW u -—4—7:w— { P157“ 1 = FL , , . “._.r ‘7 KFL‘ ' \ C i ! 'JCL C z- . M f ‘76” 4' . - ’3’ JCL'" "v 7M; -4 1-K L E _,L 2_7v(‘ “+47'57ELE '_ i “A 1 - . C 4)?“- . .l K 2" .55 "3 4‘4 C C_., 3:?" ’v" 6." .-4. {“790 fi .4 pm 4-1 V. _ 4 1/ <2 C ._. CC my 6; T f—HL "C/ J 1 ‘7L C3“ 4: V‘ C 2197(4 €26 )~._ * 4% L742“ ‘ . I: L—- i (30.3 ‘ r‘ T ,_ '3. C J /k Kgfi—C-WCU‘“) 4“? «4 690""; .+ LU ( Zr)LWF%‘> 'i‘. 4“ Q7 7“”— tj ‘F’ (ng ‘1J 7< t— 7/8 410% - ':.. '22: (2fo ” C, , ,C. - Y w," , ._. C. 4;; 'Efig A, 2.x (. .h) \r fl .0 CO ' .«fi'L, ‘H f C \_ h , . G; L MW)? '+ M; --7 / ‘ Em-rgi24 “:9 L\ /‘\ y I L L1 b1 1’ h "' {1‘ _\- (.19 L") Lia-J ‘L‘\ . 1.3, . “nu -. . a- a: 7 w ("115-‘1wé’I-3 ”2&th {VFW-"191%. ‘1‘ E}, W QM?! /[4_._ “LOC- L’ ’ :“w. v”: J "" U {“1 , V. 4 Um) 014-») mm, — -- in: ”4. ~‘U C1 am“ M Mr?) cwwemm L" W5 ) C #1 Ar 1) "j ': (a T (.31! 21 % a egg «AL—r {3110‘ + ’5% #4th . ,2 MFA} (L ( e $1713. L- ‘r‘ . ' ’X?’ “T we )40,C7(+.:S J (2:!- CO , 3 (99:2ij 7 E (Sri-i)(<j§T—}1,_S—+L)] w A. ‘E t 1 L \ [e (ii—TH 5+7531>J ,2 git—”x (lg-~13 » :2 «>< U41) 97 {29W (9 3.3 (93 L. L @ yefi )9») Cefih‘ '/ 30%] LE aim-9 9<"tg‘3\9j 1.9“» (WSW) it)” (9—9331 33 W PM 3‘: . .3, G '9- t 4. WA— 93/07 a 2 “t Z ”LT/L F U D: "3 “3” 4,65%- 04 (T """ I] 3/1”“ (5 :3 r a3 3 ~ kmu~¥f3fl<u £3 gut-v T3 . f 3115‘ V, (f; L LPG/LU '3 g_ g» 7.34:"? .. BEL: (b3 51‘ ‘ ~-- “ 9f (5) L" [ r 0- L 30 {_. (“0%)“th "(3: , "/"fi . ‘ - L L 3 3c» [0 {(Mrcwj LE». - #7 _ LR) i if! i {KW §\—: 00 ...
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