Question

Problem 5.1: Find the GS for y'"' + y" + y' + y = 1 + e* + e-* + e2x + e-2x Problem 5.2: Find the GS for y(4) – y(3) – y(2) – y(1) – 2y = 1 + x + x2 + x3

please show work, thank you!

Answer 1

IUNIE PgDn End Ins Del 5-1) gtgty'ty = lt eltek + 27 + 6 2x Auxillay equation is m3 m² +m+1=0 ma (m+1)+(m+1)=0 (mati) (m+1)=0 um=-1, I e .. Ge = ciel + G Cosx + g sinx p = 1 + et (0 + 1) (0+1) + 1) (0+1) 624) (0 +D + 1 2 + 1 e 2x 0 +1) (0+) (6+1 (D+1) ex .. General soln y = Get + Q cob x + G sinx tltext xe a To 5. d) y yo yo-y-2y = H2 + x + x² Annillary eanis mt – m3 - m - m-2=0 may is a cool > (m+1) (m3_ and fm-2)=0 Ť (M+1) (n+1) (m-2) = 0

=> m= ti tre Je=c, cost to sinnt Ge tq ex - (Hx+x+x3) Jp = 040-89-0-2 14x7x + 3 = 21- (oto3_8_D) m = I1 - 9 - Je = c, cosx + G sinx+ géte etx By method of andetermined coeff's Jp = a A+ Bx + (x² + DX3 Ipl = B + 2(x + 3D2² 9" = 2C+ 6D Spl = 200.60 gyv = 60 - 6D - 26+601) – (B+267 +30x2) - 2 (A+BX+ CX+Dx²) =Hx+x+83 -2A -B-20-60=1 2B -20 –60 = 1 -20-3D=1 -20=1

> D = 1/2 c=1+30 → C= *(1-3) =* -26 -5 -6*=) → -2B = 1+5-3 > -2B = 1/2 * 2B = 3 i ri i Alw elú det elav u I I → -2A = we + - 1 - 3 tolt >-QA = >-QA = 2 . Greneul soln is y = G COSA + $ 868t Gētcety