1. **Problem Statement:** Find the correct particular integral (PI) for the differential equation $$ (D^3 - D^2 - 6D)y = x^2 + 1 $$ where $D$ represents differentiation with respect to $x$. 2. **Understanding the Operator:** The operator is $$D^3 - D^2 - 6D = D(D^2 - D - 6)$$. 3. **Characteristic Polynomial:** The characteristic polynomial for the homogeneous part is $$r^3 - r^2 - 6r = 0$$ which factors as $$r(r^2 - r - 6) = 0$$. 4. **Roots:** Solve $$r^2 - r - 6 = 0$$ using quadratic formula: $$r = \frac{1 \pm \sqrt{1 + 24}}{2} = \frac{1 \pm 5}{2}$$ So roots are $$r = 3, -2$$ and the third root is $$r=0$$. 5. **Form of Complementary Function (CF):** Since roots are distinct, CF is $$y_c = C_1 + C_2 e^{3x} + C_3 e^{-2x}$$. 6. **Right Side (RHS) Analysis:** RHS is a polynomial $$x^2 + 1$$. 7. **Particular Integral Form:** For polynomial RHS of degree 2, try a polynomial PI of degree 3 (because of the $D$ operator multiplying the polynomial): $$y_p = Ax^3 + Bx^2 + Cx + D$$. 8. **Apply Operator to $y_p$:** Calculate derivatives: $$y_p' = 3Ax^2 + 2Bx + C$$ $$y_p'' = 6Ax + 2B$$ $$y_p''' = 6A$$ Apply operator: $$(D^3 - D^2 - 6D)y_p = y_p''' - y_p'' - 6y_p' = 6A - (6Ax + 2B) - 6(3Ax^2 + 2Bx + C)$$ Simplify: $$= 6A - 6Ax - 2B - 18Ax^2 - 12Bx - 6C$$ Group terms by powers of $x$: $$= -18Ax^2 + (-6A - 12B)x + (6A - 2B - 6C)$$ 9. **Equate to RHS:** $$-18Ax^2 + (-6A - 12B)x + (6A - 2B - 6C) = x^2 + 1$$ Match coefficients: - For $x^2$: $$-18A = 1 \Rightarrow A = -\frac{1}{18}$$ - For $x$: $$-6A - 12B = 0$$ - For constant: $$6A - 2B - 6C = 1$$ 10. **Solve for B:** $$-6(-\frac{1}{18}) - 12B = 0 \Rightarrow \frac{1}{3} - 12B = 0 \Rightarrow 12B = \frac{1}{3} \Rightarrow B = \frac{1}{36}$$ 11. **Solve for C:** $$6(-\frac{1}{18}) - 2(\frac{1}{36}) - 6C = 1$$ $$-\frac{1}{3} - \frac{1}{18} - 6C = 1$$ $$-\frac{6}{18} - \frac{1}{18} - 6C = 1$$ $$-\frac{7}{18} - 6C = 1$$ $$-6C = 1 + \frac{7}{18} = \frac{18}{18} + \frac{7}{18} = \frac{25}{18}$$ $$C = -\frac{25}{108}$$ 12. **Particular Integral:** $$y_p = -\frac{1}{18}x^3 + \frac{1}{36}x^2 - \frac{25}{108}x + D$$ Since constant $D$ can be absorbed in complementary function, omit it. 13. **Rewrite $y_p$:** Multiply numerator and denominator to match options: $$y_p = \frac{1}{6} \left(-\frac{x^3}{3} + \frac{x^2}{6} - \frac{25x}{18}\right)$$ 14. **Compare with options:** The correct particular integral matches $$-\frac{1}{6} \left( \frac{x^3}{3} - \frac{x^2}{6} + \frac{25x}{18} \right)$$ **Final answer:** $$\boxed{-\frac{1}{6} \left( \frac{x^3}{3} - \frac{x^2}{6} + \frac{25x}{18} \right)}$$