求x''+6x'+13x=(x^2-5x+2)e^x的通解

Question

常微分的问题～

Answer 1

The aux. equation
p^2+6p+13=0
(p+3)^2 = -4
p =-3/2+ i or -3/2- i
let
yg= e^(-3x/2) . [ Acosx +Bsinx ]
let
yp=(Cx^2+Dx+E).e^x
yp'
=(Cx^2+Dx+E + 2Cx +D).e^x
= [Cx^2+(2C+D)x+(D+E) ].e^x
yp''
= [Cx^2+(2C+D)x+(D+E) +2Cx + 2C+D].e^x
= [Cx^2+(4C+D)x+(2C+2D+E)].e^x
y''+6y'+13y=(x^2-5x+2)e^x
[Cx^2+(4C+D)x+(2C+2D+E)].e^x + [6Cx^2+6(2C+D)x+6(D+E) ].e^x
+(13Cx^2+13Dx+13E).e^x = (x^2-5x+2)e^x
[ 20Cx^2 +( 16C+20D)x +(2C+8D+13E) = (x^2-5x+2)e^x
C =1/20
16C+20D = -5
4/5 +20D=-5
D=-29/100
2C+8D+13E =2
1/10- 232/100 +13E=2
E=211/50
yp=[(1/20)x^2 -(29/100)x + 211/50 ].e^x
y
= yg+yp
=e^(-3x/2) . [ Acosx +Bsinx ] +[(1/20)x^2 -(29/100)x + 211/50 ].e^x本回答被网友采纳

Answer 2

非齐次方程特解为y=(1/20x²-29/100x+211/1000)e^x
再加上齐次方程的通解（这个比较好算）就是答案了