This document contains solutions to several differential equations: 1) Y = C1 + C2e3x + 4/5 e5x – 2/5 sen x + 6/5 cos x 2) Y1 = C1Cosx + C2 sen x + 1/5xcosx – 1/20x2cosx -9/20xsenx+1/5x2senx 3) Y = C1cos2x + C2 sen2x + 4/3 cosx + Sen x-2

1. Y’’ -3y´ =8e 5x+4 sen x M2 – 3m=0 m(m-3) m=0 m=3 Yc= C1 + C2 e 3X Yp=C3e 5x + c4senx+ C5cos x Yp’= C3e5x Yp’=5C3e5x Yp’’=25C3e5x 25C3e5x - 15C3e5x -C3e5x 10C3=8 C3=4/5 YP2= C4sen x + C5cos x YP2= C4cos x - C5sen x YP2= -C4sen x - C5cos x -C4sen x – C5 cos x – 3C4 Cos x + 3C5en x= 4sen x -C4 + 3c5=4 3c4+4c5=-12 -c5-3c4 =0 -3c4-c5 =0 -10c5=12 c5= 6/5 -6/5 -3 c4=0 3c4= -6/5 c4 =--6/15 =-2/3 c4= -2/5 Y=C1+C2e3x + 4/5 e5x – 2/5 sen x + 6/5 cos x Y’’ +y´ =xCosx - cos x M2 + 1= 0 M2=-1 M=Raiz -1 m= 1i Yc= C1 cosX + C2 Ssen x Yp= C3 x cosx + C4x2 cosx + C5xsenx+ C6 x2senx Yp’= C3 cosx - C3x senx +2C4xcosx- C4 x2senx-C5senx+C5 x cos x + 2C6xsenx +C6x2cosx Yp’’= -2C3senx-C3xsenx+2C4cosx-4c4xsenx-C4x3cosx 2C5senx-C5xsenx+2C6senx+4c6xcosx-C6x2senx -2C3senx – C3xsenx + 2C4cosx – 4C4xsenx-C4x2cosx + 2C5cosx –C5xsenx+2c5senx +4c6xcos –c6x2senx+C3xcosx+C4x2cosx+C5xsenx + C6x2sen x = xcosx-cosx -2C3senx – C3xsenx + 2C4cosx – 4C4xsenx-C4x2cosx + 2C5cosx – +4c6xcosx +C3xcosx+ 2C6senx = xcosx -cosx -2C3 +2C6=0 C6=C3 -C3 -4C4 = 0 2C4 +2C5=-1 5C6=1 4C6+C3=1 C4=-1/20 C5=-9/20 C6=1/5 C3=1/5 Y1= C1Cosx + C2 sen x + 1/5xcosx – 1/20x2cosx -9/20xsenx+1/5x2senx Y’’ +4y =4Cosx +3senx-8 M2+4=0 m=2i Yc= C1 cos2x + C2 sen2x Yp= C3 cosx + C4 sen x + C5 Yp’= -C3senx – C4cosx Yp’’=-C3cos x - C4sen x -C3cosx – C4sen x + 4C3cosx + 4c4senx +4c5=4Cos x + 3Sen x –x C5=-2 3C3=4 3C4=5 C4=1 C3=-4/3 Y=C1cos2x + C2 sen2x + 4/3 cosx + Sen x-2