observe that x and e^x are solutions to the homogeneous equation associated with:
(1-x)y'' + xy' - y = 2(x - 1)^2 e^-x
use this fact to solve the nonhomogeneous equation.
explain step by step and show your work, you. from left to right it's y double prime + xy

To take advantage of the characteristic solutions and , you can try the method of variation of parameters, where we look for a solution of the form

with the condition that

Then

Substituting into the ODE gives

Since

the above reduces to

and form a linear system that we can solve for using Cramer's rule:

where is the Wronskian determinant of the fundamental system and is the same determinant, but with the -th column replaced with .

So we have

Then the particular solution is

giving the general solution to the ODE,

b

step-by-step explanation: