1. The problem asks to discuss the type of the family of second order systems (SOS) in terms of the parameter $k$. 2. Typically, the type of a second order system depends on the characteristic equation, which is often of the form $$s^2 + 2\zeta\omega_n s + \omega_n^2 = 0,$$ where $\zeta$ is the damping ratio and $\omega_n$ is the natural frequency. 3. The parameter $k$ will affect the coefficients of the characteristic equation, thus changing the roots and the system type (overdamped, underdamped, critically damped, or unstable). 4. To analyze, write the characteristic equation in terms of $k$, then find the discriminant $$\Delta = b^2 - 4ac$$ of the quadratic equation. 5. If $$\Delta > 0$$, roots are real and distinct (overdamped). 6. If $$\Delta = 0$$, roots are real and repeated (critically damped). 7. If $$\Delta < 0$$, roots are complex conjugates (underdamped). 8. If any root has positive real part, the system is unstable. 9. Without the explicit equation, the general approach is to express the characteristic polynomial in terms of $k$, compute $$\Delta$$, and classify the system accordingly. 10. For example, if the characteristic equation is $$s^2 + 2ks + 1 = 0,$$ then $$a=1, b=2k, c=1,$$ and $$\Delta = (2k)^2 - 4(1)(1) = 4k^2 - 4 = 4(k^2 - 1).$$ 11. Then: - For $$k^2 > 1$$, $$\Delta > 0$$, system is overdamped. - For $$k^2 = 1$$, $$\Delta = 0$$, system is critically damped. - For $$k^2 < 1$$, $$\Delta < 0$$, system is underdamped. 12. This method can be applied to any SOS family by substituting the characteristic equation coefficients in terms of $k$.