Question

求通解第三题的第1和第3问

Answer 1

(1)
2y''+y'-y=2e^x
The aux. equation
2p^2+p-1=0
(2p-1)(p+1)=0
p=1/2 or -1
let
yg= Ae^(x/2) +Be^(-x)
let
yp = Ce^x
2yp''+yp'-yp=2e^x
2Ce^x =2e^x
C=1
yp= e^x
y=yg+yp= Ae^(x/2) +Be^(-x) + e^x
(3)
2y''+5y'=5x^2-2x-1
The aux. equation
2p^2+5p=0
p(2p+5)=0
p=0 or -5/2
yg = A+ Be^(-5/2)
let
yp = Cx^3+Dx^2+Ex+F
yp'=3Cx^2+2Dx+E
yp''=6Cx+2D
2yp''+5yp'=5x^2-2x-1
2(6Cx+2D)+5(3Cx^2+2Dx+E) = 5x^2-2x-1
15Cx^2+(12C+10D)x +4D+5E =5x^2-2x-1
15C =5 (1)
12C+10D=-2 (2)
4D+2E = -1 (3)
from (1)
C=1/3
from (2)
4+10D=-2
D=-2/5
from (3)
4D+2E = -1
-8/5+2E=-1
E = 3/10
yp = Cx^3+Dx^2+Ex+F =(1/3)x^3 -(2/5)x^2 +(3/10)x +F
y
=yg+yp
= A+ Be^(-5/2) +(1/3)x^3 -(2/5)x^2 +(3/10)x +F
=Be^(-5/2) +(1/3)x^3 -(2/5)x^2 +(3/10)x +H