Question

a. Show that y = y + y2 is a solution of y" + P(x)y' + Q(x)y = T (x) + T2(x) if y, and y, are the solution of the following equations respectively; y + P(x)y' + Q(x)y = Ti(x) and y" + P(x)y' + Q(x)y = T2(x) (CO2:P01 - 4 Marks) b. Determine the general solution of the given equation using method of undetermined coefficients y" +9y = 2 sin 3x + 4 sin x - 26e-2x + 27x3 The idea of Q 1(a) can be applied. (CO2:P01 - 16 Marks)

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Answer 1

a show Y = 4, +42 is a son of. Y"+ P(24) 4' + Qere) y = Ti(de) + T2(e). - guen Y, and Y₂ are son of the following equations respectively. y't Pra) y'+Q(4) Y = Tila) and: TECH). HP!" -22 27213_2682 let Y "+ Plansyl + Q1e) y = T2 (H) then Y, and Y₂ Satisfies the equations respectively Y," + P (24) Y! + Q(H)4, = Ti CH) f you + P(H) y + QH) Y 0 +2 Y, + Y " + PcH) (Y,'t Y₂ ) + Q (H) (Y,+Y₂) = Ticho+T2(x) (Y, TY2) " + Pese) (4, +42}'+Qc HJ LY, TY₂) = Ti (gc) + Tz (21) Ž 9, +42 is a son of the eq. b) Y"+g4=2 sin3x+usinx-2602x+ 27x3 Y"+gy = 25 in 3x + using t Tica) Tack). Then let Y" + gy asin 3xtusina and y"+gy = 2773_266-24 for. Ye m²+9 20 ti Yo = G ces 34+ siuza. f(0) = D2+9 2 sinan 4 sina D²+9 D2+9 yp =- Los(BH) + 4 sinne t m For Yo 2x (-1053x) + £ sina 2(3) + Up

Ya+Yp G cos3x + Gsin 3x + 12 surse - e cos (3H). now for yu+gy -2x = 2703_26 y, lea yot yer w you -20 272396e yer G cos (3x) + Cysin (3x). (0²+9) 27 D²+9 وهي you 2723 26, 6-24 9 [1+ $?J (-2)2 +9 .-22 ze 2 3 13 3 [1+0² 3 [1-e+ 4 + ---] x2-26-28 $3 그 3[33 - Gul to-~] - 20-22 IP2 = 303 -20 zeza ya Ycz + Yo Cz cas (32) + Cusin (34) + 3x²_28-2é 6-22 y= Y,+Y, (G+3) Cos 32c + CC tou sin 34 +323-22-24-24 t i sinne cos (3x) y = C, cos se t C sin 3x + 1 sins ICOS 3 X) +373-24-20 X 3 - 22