Question

y''-4y'+y=x^2-2x+3

Answer 1

This is fairly longish.
First you have to solve this:
y''-4y'+y=0
Do you know how to do that?

Answer 2

the auxillary function for the above given equation is D^2-4D+1 =0
solving we have for D we have
D =( 4+-sqrt(16-4))/2 = 2+- sqrt(3)
hence the CF of the equation is given by
y = e^2x (Ae^(sqrt(3)x) +Be^(-sqrt(3)x) )...

Answer 3

now we need to find the PI for x^2-2x+3
which is given by
1/(1+ D^2-4D ) * ( x^2-2x+3)
= (1+4D-D^2 +16D^2) *( x^2-2x+3)
=(1+4D+15D^2)*( x^2-2x+3)
=(( x^2-2x+3) +4(2x-2) +15*2)

Answer 4

hence complete solution is y=CF +PI