1. **State the problem:** We are given a second-order differential equation describing a system: $$20 \frac{d^2x}{dt^2} + 100 \frac{dx}{dt} + 500x = u$$ We need to find the system's time constant, damping ratio, natural frequency, and transfer function. 2. **Write the standard form of a second-order system:** The standard form is: $$\frac{d^2x}{dt^2} + 2\zeta\omega_n \frac{dx}{dt} + \omega_n^2 x = \frac{1}{m} u$$ where $\zeta$ is the damping ratio, $\omega_n$ is the natural frequency, and $m$ is the mass or coefficient of the second derivative term. 3. **Normalize the given equation:** Divide the entire equation by 20 to get the standard form: $$\cancel{20} \frac{d^2x}{dt^2} + \cancel{20} \times 5 \frac{dx}{dt} + \cancel{20} \times 25 x = \frac{1}{20} u$$ which simplifies to: $$\frac{d^2x}{dt^2} + 5 \frac{dx}{dt} + 25 x = \frac{1}{20} u$$ 4. **Identify parameters:** From the normalized equation: $$2\zeta\omega_n = 5$$ $$\omega_n^2 = 25$$ 5. **Calculate natural frequency $\omega_n$:** $$\omega_n = \sqrt{25} = 5$$ 6. **Calculate damping ratio $\zeta$:** $$2\zeta \times 5 = 5 \Rightarrow 10\zeta = 5 \Rightarrow \zeta = \frac{5}{10} = 0.5$$ 7. **Calculate time constant $\tau$:** The time constant for a second-order system is often related to the damping and natural frequency by: $$\tau = \frac{1}{\zeta \omega_n} = \frac{1}{0.5 \times 5} = \frac{1}{2.5} = 0.4$$ 8. **Find the transfer function:** Taking Laplace transform assuming zero initial conditions: $$20 s^2 X(s) + 100 s X(s) + 500 X(s) = U(s)$$ Factor out $X(s)$: $$X(s) (20 s^2 + 100 s + 500) = U(s)$$ Transfer function $G(s) = \frac{X(s)}{U(s)}$ is: $$G(s) = \frac{1}{20 s^2 + 100 s + 500}$$ Divide numerator and denominator by 20: $$G(s) = \frac{1/20}{s^2 + 5 s + 25}$$ **Final answers:** - Natural frequency $\omega_n = 5$ - Damping ratio $\zeta = 0.5$ - Time constant $\tau = 0.4$ - Transfer function $G(s) = \frac{1/20}{s^2 + 5 s + 25}$