1. **State the problem:** We have a discrete-time system described by the difference equation $$y(k+2) + 5y(k+1) + 6y(k) = U(k)$$ where $$U(k) = 1$$ for $$k \geq 0$$. We want to find the solution $$y(k)$$. 2. **Identify the type of equation:** This is a non-homogeneous linear difference equation with constant coefficients. 3. **Solve the homogeneous equation:** First solve $$y(k+2) + 5y(k+1) + 6y(k) = 0$$. The characteristic equation is $$r^2 + 5r + 6 = 0$$. 4. **Find roots:** Factor or use quadratic formula: $$r^2 + 5r + 6 = (r+2)(r+3) = 0$$ So, $$r = -2$$ or $$r = -3$$. 5. **General homogeneous solution:** $$y_h(k) = C_1(-2)^k + C_2(-3)^k$$ where $$C_1, C_2$$ are constants. 6. **Find particular solution:** Since $$U(k) = 1$$ (a constant input), try a constant particular solution $$y_p(k) = A$$. Substitute into the difference equation: $$A + 5A + 6A = 1$$ $$12A = 1$$ $$A = \frac{1}{12}$$. 7. **Complete solution:** $$y(k) = y_h(k) + y_p(k) = C_1(-2)^k + C_2(-3)^k + \frac{1}{12}$$. 8. **Initial conditions:** To find $$C_1$$ and $$C_2$$, initial conditions $$y(0)$$ and $$y(1)$$ are needed. Since not given, the solution is expressed in terms of these constants. **Final answer:** $$y(k) = C_1(-2)^k + C_2(-3)^k + \frac{1}{12}$$