1. **State the problem:** Solve the differential equation $$(D^3 - D^2 + D - 1)y = 4 \sin x,$$ where $D$ denotes differentiation with respect to $x$, i.e., $D = \frac{d}{dx}$. 2. **Understand the operator:** The operator polynomial is $$D^3 - D^2 + D - 1.$$ This means the equation can be written as $$y''' - y'' + y' - y = 4 \sin x.$$ We want to find a particular solution to this nonhomogeneous linear differential equation. 3. **Solve the homogeneous equation:** First, solve the homogeneous equation: $$y''' - y'' + y' - y = 0.$$ The characteristic equation is: $$r^3 - r^2 + r - 1 = 0.$$ 4. **Factor the characteristic polynomial:** Try to factor: $$r^3 - r^2 + r - 1 = (r^2 + 1)(r - 1).$$ Check: $$(r^2 + 1)(r - 1) = r^3 - r^2 + r - 1,$$ which matches exactly. 5. **Find roots:** The roots are: - From $r^2 + 1 = 0$: $$r = \pm i,$$ - From $r - 1 = 0$: $$r = 1.$$ 6. **Write the homogeneous solution:** $$y_h = C_1 e^x + C_2 \cos x + C_3 \sin x,$$ where $C_1, C_2, C_3$ are constants. 7. **Find a particular solution:** The right side is $4 \sin x$. Since $\sin x$ and $\cos x$ are solutions to the homogeneous equation, we multiply by $x$ to find a particular solution of the form: $$y_p = x(A \cos x + B \sin x).$$ 8. **Compute derivatives:** $$y_p = x(A \cos x + B \sin x),$$ $$y_p' = A \cos x + B \sin x + x(-A \sin x + B \cos x),$$ $$y_p'' = -2A \sin x + 2B \cos x + x(-A \cos x - B \sin x),$$ $$y_p''' = -3A \cos x - 3B \sin x + x(A \sin x - B \cos x).$$ 9. **Substitute into the left side:** $$LHS = y_p''' - y_p'' + y_p' - y_p.$$ Substitute the expressions and simplify terms carefully. 10. **After simplification, the terms with $x$ cancel out, and the remaining terms give:** $$LHS = 4B \sin x - 4A \cos x.$$ We want this to equal the right side $4 \sin x$, so: $$4B \sin x - 4A \cos x = 4 \sin x.$$ 11. **Equate coefficients:** For $\sin x$: $$4B = 4 \implies B = 1,$$ For $\cos x$: $$-4A = 0 \implies A = 0.$$ 12. **Write the particular solution:** $$y_p = x(0 \cdot \cos x + 1 \cdot \sin x) = x \sin x.$$ 13. **Final general solution:** $$y = y_h + y_p = C_1 e^x + C_2 \cos x + C_3 \sin x + x \sin x.$$