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为äºæè¿°æ¹ä¾¿ï¼è®°u=e^(-2x), åu'=-2u, u"=4u.
设y=uvï¼åy'=u'v+v'u=(v'-2v)u.
y"=(v"-2v')u+(v'-2v)u'=(v"-2v'-2v'+4v)u=(v"-4v'+4v)u.
y"+3y'+2y=(v"-4v'+4v)u+3(v'-2v)u+2uv=(v"-4v'+4v+3v'-6v+2v)u=(v"-v')u=xu, å³æ
v"-v'=(v'-v)'=x, æ以v'-v=x^2/2+C'.
å设v=ax^2+bx+c, åv'=2ax+b,
ç±v'-v=2ax+b-ax^2-bx-c=-ax^2+(2a-b)x+(b-c)=x^2/2+C'. å¾
a=-1/2, 2a-b=0ï¼æ以b=-1, céæ
æ以v=-x^2/2-x+C.
å æ¤y=(-x^2/2-x+C)e^(-2x)ï¼æ£éªä¸ä¸å
y'=-2(-x^2/2-x+C)e^(-2x)+(-x-1)e^(-2x)=(x^2+2x-2C-x-1)e^(-2x)=(x^2+x-2C-1)e^(-2x).
y"=-2(x^2+x-2C-1)e^(-2x)+(2x+1)e^(-2x)=(-2x^2-2x+4C+2+2x+1)e^(-2x)=(-2x^2+4C+3)e^(-2x).
å æ¤y"+3y'+2y=(-2x^2+4C+3)e^(-2x)+3(x^2+x-2C-1)e^(-2x)+2(-x^2/2-x+C)e^(-2x)=(-2x^2+4C+3+3x^2+3x-6C-3-x^2-2x+2C)e^(-2x)=xe^(-2x).
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