Question

解这个微分方程，谢谢

Answer 1

y''-y=4x.sinx
The aux. equation
p^2-1=0
p=1 or -1
let
yg=Ae^x +Be^(-x)
let
yp= (Cx+D)sinx +(Ex+F)cosx
yp'
=(Cx+D)cosx + Csinx -(Ex+F)sinx + Ecosx
=[-Ex +(C-F)] sinx + [Cx +(D+E)]cosx
yp''
=[-Ex +(C-F)] cosx -Esinx - [Cx +(D+E)]sinx + Ccosx
=[ -Cx +(-D-2E)]sinx +[-Ex +(2C-F)]cosx
yp''-yp=4x.sinx
[ -Cx +(-D-2E)]sinx +[-Ex +(2C-F)]cosx -(Cx+D)sinx -(Ex+F)cosx = 4x.sinx
[ -2Cx +(-2D-2E)]sinx +[-2Ex +(2C-2F)]cosx= 4x.sinx
-2C =4 (1)
-2D-2E=0 (2)
-2E=0 (3)
2C-2F=0 (4)
from (1), (3)
C=-2, E=0
from (2)
-2D-2E=0
-2D-0=0
D=0
from (4)
2C-2F=0
-4-2F=0
F=-2
yp= (Cx+D)sinx +(Ex+F)cosx
=-2x.sinx -2cosx
y= yg+yp
=Ae^x +Be^(-x) -2x.sinx -2cosx

追问

高手 你这是自己做的？还是用计算机做的？