We need to solve the second-order linear non-homogeneous differential equation: $y'' + y = x^2 - x$.

1. Problem Description

We need to solve the second-order linear non-homogeneous differential equation: $y'' + y = x^2 - x$.

2. Solution Steps

First, we find the homogeneous solution. The characteristic equation is $r^2 + 1 = 0$. The roots are $r = \pm i$. Therefore, the homogeneous solution is $y_h = c_1 \cos(x) + c_2 \sin(x)$.

Next, we find a particular solution. Since the right-hand side is a polynomial of degree 2, we assume a particular solution of the form $y_p = Ax^2 + Bx + C$. Then, $y_p' = 2Ax + B$ and $y_p'' = 2A$.

Substituting into the differential equation, we have $2A + Ax^2 + Bx + C = x^2 - x$.

Comparing coefficients, we get:

$A = 1$

$B = -1$

$2A + C = 0$, so $2(1) + C = 0$, which implies $C = -2$.

Therefore, $y_p = x^2 - x - 2$.

The general solution is the sum of the homogeneous and particular solutions: $y = y_h + y_p$.

3. Final Answer

$y = c_1 \cos(x) + c_2 \sin(x) + x^2 - x - 2$