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240-2006&amp;2007-2-M20-May2007

# 240-2006&amp;2007-2-M20-May2007 - Kuwait University...

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Unformatted text preview: Kuwait University ' Math. 240 May 01; 2607 Dept. of Math. 85 Comp‘ Sci Second Midterm Exam Duration 90 Minutes Calculators and mobile phones are not allowed Answer the following questions. i. (a) (3 points) If f (at) = me” is a solution of a second-order linear diflerential equation, ﬁnd a linearly independent second solution 9 (as) such that “LU (f: 5?) : 553 where w is the wronsldan. (b) (3 points) Let A : (I) + 1‘} (.L'D +1) and B = (3D + 1) (D —+- 3:) Show that AB % BA 2. Use inverse Operators to ﬁnd the the general solution of the following equations (IL) (4 points) (0’‘ —l-- D3 -I~ D *I— 1) y = 1+ m2 (b) (5 Points) (192 —_- 4] 3; 2 oz” cos 2;; + (josh 2:1: .-. 3. (5 Points) Find the general solution of the equation _ my" + (1 e 2.7:) y’ + (:c ~1)y’-= me“ given that y; = e” is a solution of the corresponding homogeneous equation 4. (5 Points) Use variation of parameters method to ﬁnd the general solution of the equation _ I y” + y = cots Some formulas for your beneﬁt _ e” if a —— 0 Box I 1 f(D) \$5: (1.: ( ) f = leMG) 1 f(a} O&¢(a)9éﬂ emetic) = “I [ 1 v(x)] (2) f (D) ' f(D + a) 1 sin in: 1 sin bx 2 2 D2+a2{c0\$bz _ itzQ-kiﬂlrccisln:1 a #b (3) 1 { sin ex .1: — cos om D2 + a2 O 8 o H [I MI 9 C” E‘ o H 33 E Math. 240 Second Midterm Exam Solution May 01, 2007 are” 3 (m) z (I + 1) 9’ (I) B @ grw(m+1)g=mze==~m=e-r£:—w:e~1(1+é>ds=g: 9(29) (g) = [(6%) (\$26‘”)dm= f“ —2zd\$__1(gx+1)e_2m mag - - * ----— (b) (D+:r)(wD+1)y=(D+a:)( )(ny+y)= Dy+xD2y+Dy+\$2Dy+my A=mD?+ +(a;2 +2) 0+2: @ (rD+1)(D+2:)y=(wD+1)(Dy+:cy)=mD2y+m(y+\$Dy)+Dy+scy B=xD2+(\$2-I—1)D+2x =>A5£[email protected] ' 1. (a)(z'=)W =1t:3=>awfﬂq’—ge”(:1:+1)=13 2' (a) y—yc+yp ' m4 +m3 +m+1=0=:>(m+1}(m3 +1)==‘;‘((1'1".:+1)2(1n'r12 —m+1)=0 =>m= —1, —1, %:|:32£1' ya: (clx+r:2)e7”+e”/2 (63C08%\$+C4S‘1HJ§33) @ 1 2 2' yPZWUHFU—WDW”)=_ (92: (b) y= yc+yp1 +3119: =0==r~m— 222—»— CD 1 ----- 1 _ e2”(D+1) 7 ym— —)--”~*-—€2 ”cos2x- D(D+ _4————::cos2a:— :(Dz—_1)cos2myl, I wwnwszz=_ @ mainh2a: - gm = p2_4°°5h2”=m”:m @ 3' y = 91” = 83”, y’ = ex,” + 8:11,, y" = 831) + 281.”! + 6.1“” 1 . c v”+—1f=1,@ v’=%x+;1, U=i\$2+clhllﬂll+02 y=e”(1:c 2—1—c11n|:::'|+02) Ci) 4. yc=clcosx+c2sin\$, yp:u1cosx+ugsinm (D 113(cosrc) +u’23insc :0 11'1 =—¢os:1: n i '7 I I l u’1(~smm)+u2cos:c =cotm® u'z =cscas—31na: ”1 hm =smccln users—cots: {11.2 =1n|cscm—cot:n|+cosm 9" I ‘ ® CO ...
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240-2006&amp;2007-2-M20-May2007 - Kuwait University...

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