i tried finding the second linear independant solution and have got close to it i believe how ever i still have some stuff in the way which i can seem to get rid of , and also i was wondering how would i also show with evidence that this is the correct second independent solution. 
$$(t-1)x''(t) - (2t-1)x'(t) + 2x(t) = (t-1)^2$$
$$x(2) = 0 \quad \quad \text { and } \quad\quad x'(2) = 1$$
where you are given that one independent solution to the homogeneous equation is $c^{2t}$
$(a)$ Use the method of reduction of order to show that the other independent solution is $2t-1.$