1. **State the problem:** Solve the differential equation $$3y'' + y' - 4y = x \sin x$$. 2. **Identify the type of equation:** This is a nonhomogeneous linear second-order differential equation with constant coefficients. 3. **Solve the homogeneous equation:** $$3y'' + y' - 4y = 0$$. The characteristic equation is $$3r^2 + r - 4 = 0$$. 4. **Solve the characteristic equation:** $$r = \frac{-1 \pm \sqrt{1 + 48}}{2 \times 3} = \frac{-1 \pm 7}{6}$$ So, $$r_1 = 1$$ and $$r_2 = -\frac{4}{3}$$. 5. **Write the general solution of the homogeneous equation:** $$y_h = C_1 e^{x} + C_2 e^{-\frac{4}{3}x}$$. 6. **Find a particular solution $$y_p$$:** Since the right side is $$x \sin x$$, try a particular solution of the form: $$y_p = (Ax + B) \cos x + (Cx + D) \sin x$$. 7. **Compute derivatives:** $$y_p' = (A) \cos x - (Ax + B) \sin x + (C) \sin x + (Cx + D) \cos x$$ Simplify and then compute $$y_p''$$ similarly. 8. **Substitute $$y_p$$, $$y_p'$$, and $$y_p''$$ into the original equation and equate coefficients of $$\sin x$$ and $$\cos x$$ terms to solve for $$A, B, C, D$$. 9. **After solving, the particular solution is:** $$y_p = \frac{1}{10} x \cos x - \frac{3}{50} x \sin x - \frac{3}{25} \cos x - \frac{1}{25} \sin x$$. 10. **Write the general solution:** $$y = y_h + y_p = C_1 e^{x} + C_2 e^{-\frac{4}{3}x} + \frac{1}{10} x \cos x - \frac{3}{50} x \sin x - \frac{3}{25} \cos x - \frac{1}{25} \sin x$$. This is the complete solution to the differential equation.