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240-2007&amp;2008-1-F10-Jan2008

# 240-2007&amp;2008-1-F10-Jan2008 - MATHEMATICS 8.5...

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Unformatted text preview: MATHEMATICS 8.5 COMPUTER KUWAIT UNIVERSITY SCIENCE DEPARTMENT Differential Equations Date: January 12, 2008 Answer all questions Math-240 Each question 5 marks Calculators are not allowed Final Examination Duration : 2 Hours . Solve the equation, (y—m—l)dn—(y+x—i)dy=0. . Find the general solution of the equation, ydm—m (1 + myé) dy : U. . Find the general solution of the following equation, with the aid of inverse diﬁ'erential operators formulas, (D2 7 4D + 5x202 — 8D — 1);; = 222 (92:2 + 11 sin e). . Use the method of variation of parameter to ﬁnd a particular solution of the equation, we +1)y”+(2 , m2)?! , (2 + 2):: = (z + if, given y1 = eI and y; = :12" are solution of corresponding homogenous equation. 1 t . Prove that if L‘1{f{s)} = F{t), then L‘il {ﬂkaj} : EF(E)’ k > U. . Express H (t) in terms of step function, then solve the following initial value problem using Laplace transform, X"(t) + X05) HOE); X(0) = 1, X’(0) = 0, 3 ;05t<_14, and Hm = {Qt—5 ; t>4. . Find ﬁrst ﬁve nonzero terms of power series solutions of the following differential equation near origin. State the recurrence relation and specify the radius of convergence, y”+(wrlly’—3y=0- (3.) Verify, if the family of curves r2 = r'ce’zl'yg and the family of curves x2 +— ay2 = 1 are orthogonal trajectories of each other. (in) Evaluate if1 {m} . (c) State whether the equation is linear or nonlinear and give its order, 3;" + 2 (1,03 — 8y = :92 + cos mi SEE THE BACK or THE PAGE FOR FORMULAS =>=>:> . i \ 90 DIFFERENTIAL EQUATIONS (MATH 240) I INVERSE OPERATOR FORMULA LAPLACE TRANSFORM FORMULA 14F t» = r = 3 ( (s) qun — sf(s) — Fm) L{F"(t)} ms) — am) — mu) k:n— wwm = suns) — z 1 stF‘Bm) _ Law—a» = M0) ' L“1{e‘°’f(s)}= an — cw — c) = U i L—1{f(s)g(s)} Mam—man n— OO 99 3L Solve the equation. (y—X—1)dx«- (y+ X—1)dy= 0. Sol". dy y—X_1 y—X—1=0 X=ﬂ —: SO XEO: =1 Let dx y+X—] y+X_1=0 Jr’ y=v+1® So %= :13 (homogenous) Let V=mdv=zdu+udz (1:) ﬂ: Liz—u: z_—l dz_ 22+l 2+1 1d z+udﬂ ”2+1! 2+] dud—U 2+1: Zz—+ldz=— U dﬂ (D _ _ 2 tan—12+ %ln(22+ 1) = —lnu+inC; a Huffy—XI) +]n[xi1 + (yxl) tan—'(y;1)+ln[1ix2+(yil)2]=lnC @ % Find the general solution of the equation. ydx- x(1+ dey= o. :|=lnC Sol”? Equation is Bernoulli's in x ﬂ——X=)r2y3, :>r2dx— 1r1=y3, Let z=x’l, dz=—X'2dx ® dy y dy y _ﬂ_l = E i =_ ‘dJ’ dy yz ,:.‘~ a‘y + yz )9 Linear and integrating factor #0!) = ail J” =y Q) Q = _ : __ _ = __ dy+z y“, =yz 5y§+CL=X 5y‘+C' ® g Use the inverse operator formula to ﬁnd the general solution of the equation (DZ—40+ 5)(2D2—SD— 1)y= e2‘(9X2+llsinX) ----- (1) Sol" (Dz—4D+5)(2132—8D— Dy: 0 ----- (2) (LE:2 ~4m+ 5)(2m2 n Sail—1): 0 Roots are (2 i 1; 2 i if) = 621(01cosx-r C2 sin X) + 660735)] + C‘4e(2+%)}'r ‘59 To ﬁnd yp yp= —W(DZ 4D+5)(ZDZ 8D—I)ostz"(§‘){2+llsinx) 15:62 y”: 82X(Dz+1)(2DZ—9) _ Jr #2... # yp‘ezizuka—V (D1+1)(2D2—9)Smx:i _ x _ 11 #[Qngmxu(D2+1)(2(_1)_ 901“] p921: (X2 +- 1J— 9)— goosx] €33 WGE + 11 sinx) (9X2 + 11 sinX) CO 9i Use the method of variation of parameter to find the particular solution of the equation X(X+ 1))” + (2 —x2))/ — (2+X)y= (x+ 1):, - - - - (1) given y; = e‘ and y: = x" are solution of corresponding homogenous equation. Sol“ Let yp = A(X)e‘+ B(X)X_l - - - - (2) 6’ X‘1 2 mylsy2)(x) : —%(1 +305", A(x)=~j “£15300”: —_[W(x+1)2dx =fx(x+l)c’d¥— ex— xarmzrsr (59 _ ex B(x)—j Wl‘wﬁuak j _12(1+X)6X(x+1)2dx =—jx2(x+1)dx= —?x3w%x“ @ Substitute in (2) yp= e‘(ex—X€'+X261)+X"(——;-X3-%X4): 621(1~X+X2)— (%x2+ if) o5. Prove that If :6 {1(5)} =m9, then 33‘ {1113)}— EH 1r> o. k)! Sol” 1(5) : Ufa—“HOdl; so that £13) 1 fe‘h’HtMt o a Let z=kg mm? dt=idz, r=0=>z— Oandt—rwﬁz-aoo fiks)=le‘”ﬁti )‘dz— 1:6 {HZ)}=:B {tks)}=—n—1§)[email protected] (1—6. Express Htr) in terms of step function, then solve the following initial value problem using Laplace transform XV?) +X(t) — 11(3'X(0)=L X'(0) = 0, 3 ;05t54, and t: HO {Zr—5 ; t>4. So!” Hrr)=3 3a(t— 4)+a(t— 4)(2r— 5)=3+2a(t— 4)(_r— 4) a) Let£{X(t)}= xrs) \$9110} + 33mg} = £{3 +2 (10— 4)(r— 4)} sins) — sX(0)—X(0) ”(5) : g + 15-45 (D 5.2 (52+ 1M5) = 5+ 1+ "35“?“ 3 2 5 ”(Shaun 43+1)+52(52+1)64 ® 2:41- 43 X“) (.s2+1)+3(g (5211))”(52 (5211)) _1 ‘1 8—45 71 6—43 xtr)=3=f 1% } 2‘13 {(a+1)}+2(£ 32 }”f’ {wimp x(t) = 3 —2cost+ 2a(t— 4)[(t— 4) — sin(t— 4)] @ . E \ E Find ﬁrst five nonzero terms of power series solutions of the following differential equation near origin. State the recurrence relation and guaranteed radius of convergence, y"+(x—1)y'—3y=0, ..... (1) Sol" Since the coefficients of y“ is constant (one) then the power series solution is vatid for all; Lat y = 23M", y = Znanx"‘, 1 y” = Eng-1A ])a"xn—2 (D n=G Substitute In (1) 211(11— 1)a,,;r"‘2 + (x~ DZnaﬂxH— 32mg" = 0 ":0 "=0 2:101 — l)a,,x“‘2 — Znanx’H + 201 — 3)a,,x” = 0 "=2 "-1 "=0 Replace n by n+2 in the ﬁrst sum and replace :1 by n+1 in the second sum i0: + 2)(n + Hangar” e i0: + l)a,,+1x" + £0: — 3)a,,x" = F1 "=0 "=0 iﬁn + 2)(n +1)a,,+2 — (n +1)a,,+.+(n — 3)a,.]x” = 0 W0 Equating the coefficients of x" to zeros for all n, we get the recurrence retation (n+2)(n+1)an+g—(n+])a,,+1+(n— 3)a,, = 0, n 2 0 I -ma > “”22 (n+2) 9"” (n+1)(n+2) ” ~ 0 63 n=0=21> a2=%a1+%ao=%(ai+3ao) n=12 a3=%a2+%a1=%(ai+a0) n = 2 2 a4 2 i(13+ L.112 = %(a1+ao)+ 21—4(a1+3ao)= 11—2(2a1+3ao) 4 12 Substitute y = Zanx" = on +aur+ogx2 +I:13;r3 +a4x“ + ----- n=Cl -au+a1x+—(a1+3ao)x2(a1+ao)x3+1—]2(2a;+3ao)x4+----- y: a(1+%x2+éx3+1x—+o-~)+a}(x+ix2+1x3+1x4+”"') 4 2 2 6 OO 99 g a) Verify, if the family of curves 2:2 = kexzﬂl and the family of curves ac2 + my2 = are orthogonal trajectories of each other. b) Evaluate £_l{—(;2-}~1—)3}. c) State whether the equation is linear or nonlinear and give its order, y” + 20/)3 — 8y = x2 + cosx. Soln a) Twofamiliesareorthogonal if (92) x(ﬂ) =_; d" m 5; (2) a5} 2: x2+y2 = 2—:[2+2 _2 _ —x22 = W 21—172 x ice =>k xe{ y)2>[2x x(2x+2ydx):|e(”’) natal” W 2 2_ -1—x2 y2(—2x)—2yy’(1—x2)_ (ﬂ) __ xy x+ay_ Z”!— J22 3 y4 _0’3 dx (2)_ 1—x2 {—}{(—)()}I [0050‘ — 2,3) — cos I} dﬁ = %[—% sin(t — 2m — ﬁcosr]; —;~ sin(t) — tcost + ~5— sint] = %[sin(r) — roost] @ c) The equation y" + 2003 m 8y : x2 + cosx is nonlinear ordinary differential equation (D of second order. 00 CO ...
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240-2007&amp;2008-1-F10-Jan2008 - MATHEMATICS 8.5...

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