1. State the problem: Solve the second-order linear differential equation $$x'' - 2x' + x = 2$$ where $x''$ is the second derivative of $x$ with respect to $t$, and $x'$ is the first derivative. 2. Find the complementary solution ($x_c$) by solving the homogeneous equation $$x'' - 2x' + x = 0$$. 3. Write the characteristic equation: $$r^2 - 2r + 1 = 0$$. 4. Factorize or use the quadratic formula to solve: $$(r - 1)^2 = 0$$ so root $r = 1$ with multiplicity 2. 5. Hence, the complementary solution is $$x_c = (C_1 + C_2 t) e^{t}$$ where $C_1, C_2$ are constants. 6. Find a particular solution ($x_p$) for the non-homogeneous equation. Since the right side is constant 2, try a constant particular solution $x_p = A$. 7. Substitute $x_p=A$ into the differential equation: $$0 - 0 + A = 2$$ which gives $$A=2$$. 8. Write the general solution: $$x = x_c + x_p = (C_1 + C_2 t) e^{t} + 2$$. 9. This is the solution to the original differential equation.