Question

3. Using undetermined coefficients / annihilator or variation of parameter and Cauchy to solve the following: (40 pts) a) 3y"- y"+ 2y'-9y = 130e2+ - 18x² +5 (10 pts)

Answer 1

3 oy Valiuta using indetermined Coetbicients/annini lator an of parameter and Cauchy to solve the following ® 3y" –y" +2y! –qy = 130222_18x*+5 sol 34"-y" + ey'-ay= 130232 -183*+5-0 let y"=r3y iylarly, glary differential equationis The homogeneous 34"y"+2y'-94= 0 3834 -8²y+zry-qy=0 (373 – y +EY-90=0 The aunil aunilary equation is. 383 r²+ 2r-q=0 r=1-3957 r=-0.531 E1-36641 The complementary solution Y !X)= c e +(CZ COSC 1.3664x) + zsin(1-3664x)) e 1.39572 -0.53117 y (2)=ce 1.3957% +(62 CUSC1:36642) +G SINC 1.3664x)) @ 0.5311X particular integrail (CⓇ) particulat solution Upx) The particulas solution for undetermined coefficient method yprx) = A**+Bx+cX+D

yox 2X 2A et 2BX+C p" 2) = 448+&B yp (X) = 81 82X (2 3. yn y +28 -94p= 130227_18x+5 - u rasi yp62); y'p(2), 8pm) values substistate in equation & 3131 627) - ( 492294 28] + 2(24e29+23x+c) –91424 BxZ++D] 130e22 - 18x2+5 27 24A 2X _ GARX _ 2B + UHRX + 4B2+ 2C- que a 130227 – 18**+5 151642-9822_ qcx +48X – 26+26-9D = 130428_ 18x345 Coefficient of e e2x coelbicient of 2² Coefficient if a -93 -18 -90 +48 = 0 B = 18 -2 15A = 130 26 cho 153 26 3. 20 -90=-4B B=2 c= 48 = 9 4(2)=8 9 19 9 : Coefficient of conslemt -2B+ 2C-9D=5 >> – 2(2)+2013)-90=5 -4-5+lry = 90. --81 +16 9 - 90 D = 65 q: A = 26 B = 2, D = - 65 9. í A, B, C, D values substistute in etuation (2) 26 Yp' 26 **+ 2x3+x-69

The pallimlar solutionis Yprx) 26 2x +2x+8X- 9 65 81 general solution is y (x)= y(x) + upra) -0.531100 1.3957% yix) = ge +((2 COS(1-36642) +3 sinc 1.36648)) e 2x e + 2x 65 81 1.39572 -0.53112 y(x) = ge +62 Cos(1-3664%)+(3 SinC1-3664X)) e tecné + 2x2+2x–55 81