Using reduction of order (conceptual) problems

[fufufu]

Homework Statement

I have a nonhomogeneous 2nd order DE and I am given one solution to it and told to use variation of parameters?
to get the general solution. First i need to get the 2nd solution though. Should I use reduction of order to get the 2nd solution? If so, then in what form should the equation be in before i start using reduction of order? The complementary solution is the solution to a homogeneous equation right? So should g(t) = 0 before using reduction of order?
heres what I mean..
the equation is:

(t-1)^2(y'') - 4(t-1)y' + 6y = t

so in order to use reduction of order should i write the equation as:
(t-1)^2(y'') - 4(t-1)y' + 6y = 0

and then proceed?

also, do i first need to divide through with the coefficient of y'' or does that matter?

Sorry if I am explaining this sort of retardedlike but I am struggling with it..
pleas help...thanks!

( fyi, the given solution is y_1 = (t-1)^2 )

Homework Equations

The Attempt at a Solution

 

[lippyka]

Yes, you should write it as (t-1)^2(y'') - 4(t-1)y' + 6y = 0 before using reduction of order. You don't need to divide through with the coefficient of y''. The complementary solution is the solution to a homogeneous equation, so yes, g(t)=0 before using reduction of order.