hw3_3_ans - Homework Assignment 7 Math 2171 - Differential...

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Unformatted text preview: Homework Assignment 7 Math 2171 - Differential Equations Sections 002 & 007, Spring 2008 Name: Due: Monday, 03/17/2008 (15pts}l. In each of the following five problems, the six roots of the characteristic equation of a certain 6-th order homogeneous differential equation with constant coefficients are given. Write the general solution of the homogeneous differential equation. (a) Roots: -1, 0, 0, 3, 3, 3 General solution: cIe‘x+ (02 +63x) + (c4 +c5x +593)?" (b) Roots: i, i, i, —i, -i, —i General solution: (c1 + czx +c3x2)cos(x) + (c4 + c5x+ c6x2)sin(x) (C)ROOtS:0,0,fi,-\/?,2+\/—§.i,2-\/-3—i General solution: 61+ 02x: +6365" + Cf'fix + e2x(C5C0S(J-5_x)+cbsin(«/Tx)) (d) Roots: 2, 2, —2, —2, 1 + [3", 1 4'5— 1 General solution: (cl+ czx)e2" + (c3+ c4024" + C53” 7‘5)" + 06514—5)" (e) Roots: 3i, —3i, 1 + 2i, 1+ 2i, 1 - 2i, 1 — 2i General solution: (c1c0s(3x) +c2sin(3x)) +ex( (c3 + 9x)cos(2x) + (c5 + ch) sin(2x)) {was}; (P11, page 180) Find the general solution of y(4)- 3313M 16y" = . r4 ~8r3 +162? = 0 r2(r2 —-8r +16) = 0 r2(r —— 4 )2 = 0 r = = 0, = = 4 Erwzwfiwsz 7’ We.“ 4x 3. GS= W) = cl + 02* + (week y.»W,~.m,mmm—~mm_m.m MM‘HW,_.‘~WW.. Wm ‘ w-WM.M.,.W (motels. (P15, page 180) Find the general solution of y“) - 8y"+ 16y =0. r4 — 8r2 +16 = 0 ' (1’2 -4)2 = 0 (r-2)2(r+2)2=0 "1 =r2:3::e::g::iwwwe mmmmmmmmmmmmmmmmmmmm 22., GS: [y(x) = (61+ c2 'x)e2" + (c3+ c4 'x)e‘2" J wwwmwnvmmnuwwemuenu.w...ew..wflH. wwwfl.” ~ (iepeeyt. (Pl 7, page 180) Find the general solution of 6y(4)+11y" + 4y = 0. 6r“ +11 r2 +4 = 0 Lets = r2. Then 6s2 +11s +4 = 0 A = 112~ 4(6)(4) = 121—16 : 25 = 52 2 . _-11—5=_i zt‘i: 2. z 23. 221?. 2.21.3— s1 2(6)' “ 3 ,wehaver 3 [m3 1] =7) 3 1,r2 3 z 2 2 2111112-; 2:2: .1... 2 f5- :JZ. .._._f_2_‘_ s2 2(6) 2,wehaver 2 [2 1] r3 21,7'4 21 gfl"m,,..W...“e"me-.m-~---..-——~.~....fiwmmm WWWWWWWWW wwwwflw—uWWwv-M’Wmulmwwm WWW m GS:ty(x) = (clcos( Z—‘lg—x)+625in(g‘/g—x)) + (03cos( gx)+c4sin(izzx)) M W uu—WWM 0. V (Emmi (1’19, page 180) Find the general solution of y(3) + y" - y - y = r3 +r2 -r -1 =0 ==> r2(r +1) - (r +1) =0 => (r +1) (r2-1)=0 (r +1)(r +1)(re1)) =0 r1 3 r2 2M; :7, 1&3.,L.WMMWW. . GS: y(x) == (61+ 6211):: "‘ +££W (iSptsfi. (P23, page 180) Solve the initial value problem y" - 6y' + 25y = 0, y(0) = 3, y'(0) = l. r2 - 6r +25 = 0 A = («6)2— 4(1)(25) = 36—100= -64 == (81')? _ 6 —~ 8 i ___ 6 +8 i = . — 2 2 3 +4 1 GS: y(x) =e3x(c1cos(4x)+czsin(4x)) 31(0): 30(01 -cos(0)+ cz-sz'n(0)) = c] = 3 y'(x) = 3 te3x(c1 ~cos(4x)+ cz-sz’n(4x)) +e3x( -4c, -sin(4x)+ 4cz-cos(4x)) y'(0) =3 'e0(cl ecos(0)+ 02~sin(0)) +e0( -4cI 'sin(0)+ 4c2'cos(0) )= 301 +402 = I =>9+462=1 =62 ~2 PS.- y(x) = e”(3c0s(4;)m— 2sin(4 ) 1 WWMWMWJ" MW“ (lSptésfl. (PZZS, page 180) Solve the initial value problem 3y(3) + 2y" = , y(0) = -1, y'(0) = 0, y"(0) = 1. 3r +2 r = 0 r2(3 r +2) = 0 2 :2: =0, : ——-—-- r1 r2 "3 3 x ; =+.+-‘g GS y(x) cl czx c3 e 2 . 2 m" y(x) =02- 3-03? 3 « =_2_._..2. .731 . y(x) 3 3039 gt: e UsingtheICs 4 " 2—.— := :2 y(0) 9031303 4 . 2 Y(0)=cz—?c3==0 => czxvc3=-- =-~ W w... -x 7” =12 .3. 2 a . PS./y(x) 4+2x+4e / . .... MW“... _ “MWW (15pt5)8. (P37, page 180) Solve the initial value problem y(4)(x) = y(3>(x), y(0) = 18, y'(0) = 12, y"(0) = 13, y(3)(0)= 7. Wm 43%) = 0 #*g=0 r3(r — I) = 0 r1=r2=r3=0, r4=1 V GS: y(x) =c1+c2x +c3x2+c4ex y'(x) == (:2 +2 63-): + c4'ex y"(x) = 2 03+ c4 -ex y{3)(x) = (34.6" Using the ICs Wm) = c4: 7 y"(0) =2c3+c4==13 ==>2c3=13 -7 =#c3=3 y'(0) =02 +c4==12 ==>c2=12 —7=5 y(0):£,+c,= 18 W 1 PS:/fv(x) = 11+ 5x +3x2 + 72‘ T 5 WWWMWMM—v ...
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hw3_3_ans - Homework Assignment 7 Math 2171 - Differential...

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