Find the general solution of the differential equation y '' - 6y '+ 8y = Xe to the power of 2x

Find the general solution of the differential equation y '' - 6y '+ 8y = Xe to the power of 2x

r^2-6r+8=0 r=2 or r=4
Homogeneous equation general C1E ^ (2x) + c2e ^ (4x)
In Xe ^ (2x), 2 is the solution of the characteristic equation
y=x(ax+b)e^(2x)=(ax^2+bx)e^(2x)
y'=(2ax+b)e^(2x)+2(ax^2+bx)e^(2x)=(2ax^2+(2a+2b)x+b)e^(2x)
y''=(4ax+2a+2b)e^(2x)+2(2ax^2+(2a+2b)x+b)e^(2x)
y''=(4ax^2+(8a+4b)x+(2a+3b))e^(2x)
Substituting into the original equation: 2x of Y '' - 6y '+ 8y = Xe
（4ax^2+(8a+4b)x+(2a+3b))-6(2ax^2+(2a+2b)x+b)+8(ax^2+bx)=x
(4a-12a+8a)x^2+(8a+4b-12a-12b+8b)x+(2a+3b-6b)=x
(-4a)x+(2a-3b)=x
-4a=1 a=-1/4 2a-3b=0 b=2a/3 =-1/6
So the special solution is: y * = (- 1 / 4x ^ 2-1 / 6x) e ^ (2x)
The general solution is: C1E ^ (2x) + c2e ^ (4x) + (- 1 / 4x ^ 2-1 / 6x) e ^ (2x)