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240-2006&amp;2007-2-F10-June2007

240-2006&amp;2007-2-F10-June2007 - Math 240 Final...

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Unformatted text preview: Math 240 Final Examination Kuwait University Math. & Comp. Sci. Dept. I Calculators and Mobile Phones are not allowed. Mon: 4 June, 2007 11:00 a.m. - 1:00 p.m. Answer all questions. Maximum Marks=40; each question is worth 5 points. 1. Find the general solution of the differential equation: y(y+3x+2)dx+x(y+x+ Udy = 0. 2. Find the general solution: 00’ _ 2 a —x(x +2y+1). 3. Find a particular solution of the non-homogeneous equation: ﬂnx yii_%yl+_x22_y= —x—’ x>0, . given two solutions y1 = x and y; = x2 of the homogeneous equation. 4. If y1 = e" is a solution of the homogeneous differential equation: I xy”+(1—2x)y'+(x—l)y=0, ﬁnd the general solution ofthe noun-hemmeous equation: ,, ,. _._ . xy"+(l—2x)y’+(x—l)y=xe". _ 5.'(a) LetL{F(r)} = ﬁs). 1r ma) = sinht, ﬁnd ﬂs). (b) Find L“ {53:1 }. 6. Use Laplace transform method to solve the initial-value problem: y"(t) + 9y(t) = F0), y(0) = 0-, y'CO) a I, where F(t) = { 3t,05t<1, 3, t>1. 7. By Laplace transform method, solve the following integral equation for F(t): Fm + £0 — 3) F03) d5 = sin(2t). 8. Find the power series solution about x = 0 for the differential equation: (x2+l)y”—2xy'+2y=0. .See'back'side for some formulas up» MN DIFFERENTIAL EQUATIONS (MATH 240) ' INVERSE OPERATOR. FORMULA _i 1 am _ .x 1 1' F[D){e 9') _ ea F(D+a)y r | W EXPONENT‘IAL SHIFT _L .J 1. emf)» = ND _ GHEMM ,— _J LAPLACE TRANSFORM FORMULA 1. mm} = m 2. ;L{F’(t)} : sf{s) — F(O) ‘ 3. L{F”(t)} = 5mg) i 5F 0) — mo) —1 k=n71 _I 4- L{F(”)(t)} = s”f(s) k E s”""‘1F“‘](0) k=0 “1‘3 . 1 o. gyms) = Jam—amt» V INVERSE LAPLACE TRANSFORM 1. ITil{f(5 —— 42)} = e“"F(t) 2. L‘1{e‘”f(s)} = (1(1,‘ — (2)1705 — c) 3. . L’1{f(s)9(s)} = jamcu u mam 4. Hi} =1 5. L-1{S..L.} = 6- L_1{.sr]+1} : 1"(:::~1)’ I > T1 7. L’1{s‘%} = ﬂ ﬂ ﬂL_1{H;a}: —at OO 99 .-J'"]A'Tj-‘!I juiugéd) 2007 Q sauna», 5' '11)” M B-rx-H ling!- N - ,1. _ a g .5; ”)#k<s+x+4>‘x 4 IF e 3‘ ® '55 s? (5'30:— 3x‘3 +u3} dxA- My +13+XZJJ3==0 F,‘ 2: 32x+3x13+23¢5 gay Fa %x191+ x35+xlfj+ C(11) F3 1-: 333+- x3+xz+ (at/(H): 313+¥5+ x1“? C”?y}=a # (“Ohm Fe: tin-x1324- (xhkﬂﬁ Au! 5'9!“ F: 9-- @ W 2.. ' _ ‘ M? 3'; ~2x3=.x(x‘~++) =9 IF-g J M" g 5*“ @ I: 'X" .. 2. Q , . Mm He! :fx(x"+f)exﬁlx+c%§ fkaJEféhcﬁﬂia+wéic tel"- ‘9 3+1+§jxl==c¢_ @ .3. Qt 35“): AC")- 2+3“), 3‘2; wgl k x1,=x\$ ----- -- ths {or Bl : r , X 1 f“: “£335 7‘) H-fxlz"j§&!~x& a": 12x St? Have", U": W, Iran 431‘s;er D5743: w:® w’+ {tear-:4 ,5, pain: @ John. .2 flxdk-P , z +5; :9 w:%,}+§ l 2. u Wﬁ-zi-X—a—qﬁm-c} \$3: (if-+6, fnx+cz)e"’ @ 45'!“ Takin L1" 3 ‘V: W \$W%\$¢ ==” Mbi-ffﬁt wﬁ’wjdﬁa-gmw (1r) -1 51H 2 ;f‘ 4‘ “31's; é". Git—J:Btutﬂd-B'ﬂwJ-allhr). 72km 1-4;” [‘H‘iywjzl‘ma ﬁu-{sJ-s.o-4qrfms)~.~._§_ 4, w: a I ' - “ ,,( e H (r) ﬂ+m Uses). 53-9 u ' --_._L___ .. . - — =7 “hm +4; 312- ;;:._; )o-e’) @ L'U. ' 9:?) séﬁfﬂﬂ+f§k ‘a‘ﬁ‘ (Bi-H‘éi'ﬁ’“ 3”“! WM) ‘ £915.: @ W!)+3‘.yum a 3;.” 5.54 g pk): 352 @ ‘4- i # a“): :95 + 3 1-1 @- !)[3 4) 3H; *4 Fftazw§5ﬁa€t\$§1s¥nat¢ @ 3. at 5.55 M on and» *? in-(hthx 2+££ﬁﬁ1fﬁ~1n+athﬁ=o @ ‘) g; ("+1)(h+’)dﬁ+.z +(ntjy+gjdﬁj.gh => a, z .. (n-Iiém +1 g a”; ”&a5-. @ "=0? 4: = a“ “do, ﬂnzaﬁrﬁ» 3', Wu!) 30:) = 4. (Mug-M, x - OO 99 ...
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240-2006&amp;2007-2-F10-June2007 - Math 240 Final...

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