求微分方程的通解.y''-6y'+9y=6e^3x

特征方程为t^2-6t+9=0,t=3
所以通解为y1=(C1x+C2)e^(3x)
设特解y2=Ax^2e^(3x)
则y2'=(3Ax^2+2Ax)e^(3x)
y2''=(9Ax^2+6Ax+6Ax+2A)e^(3x)=(9Ax^2+12Ax+2A)e^(3x)
所以9Ax^2+12Ax+2A-18Ax^2-12Ax+9Ax^2=6
A=3
所以y=y1+y2=(3x^2+C1x+C2)e^(3x)