# Stability Analysis of the Given ODEs To determine the stability of each system, **find the characteristic equation**, solve for its roots, and analyze their real parts: - **Stable:** All roots have negative real parts. - **Unstable:** Any root has a positive real part. - **Marginally stable:** Roots have zero real parts, but none repeated (except simple poles at zero). - **Neutrally stable:** All roots have zero real parts, but may have repeated zeros or pure imaginary parts. --- ## (a) \( \ddot{x}(t) + 9\dot{x}(t) + 14x(t) = 10u(t) \) ### Characteristic Equation: \[ s^2 + 9s + 14 = 0 \] ### Roots: \[ s = \frac{-9 \pm \sqrt{81 - 56}}{2} = \frac{-9 \pm 5}{2} \] \[ s_1 = \frac{-9 + 5}{2} = -2, \quad s_2 = \frac{-9 - 5}{2} = -7 \] ### Conclusion: - Both roots have **negative real parts**. - **System is Stable.** --- ## (b) \( \ddot{x}(t) + 48\dot{x}(t) + 16x(t) = 0 \) ### Characteristic Equation: \[ s^2 + 48s + 16 = 0 \] \[ s = \frac{-48 \pm \sqrt{48^2 - 4 \cdot 16}}{2} \] \[ s = \frac{-48 \pm \sqrt{2304 - 64}}{2} = \frac{-48 \pm \sqrt{2240}}{2} \] \[ \sqrt{2240} \approx 47.4 \] \[ s_1 \approx \frac{-48 + 47.4}{2} = -0.3, \quad s_2 \approx \frac{-48 - 47.4}{2} = -47.7 \] ### Conclusion: - Both roots have **negative real parts**. - **System is Stable.** --- ## (c) \( 4\dot{x}(t) - 4x(t) = f(t) \) ### Homogeneous Equation: \[ 4\dot{x}(t) - 4x(t) = 0 \implies \dot{x}(t) - x(t) = 0 \] ### Characteristic Equation: \[ s - 1 = 0 \implies s = 1 \] ### Conclusion: - **Positive real part** root. - **System is Unstable.** --- ## (d) \( \dot{x}(t) + 2x(t) = f(t) \) ### Homogeneous Equation: \[ \dot{x}(t) + 2x(t) = 0 \] ### Characteristic Equation: \[ s + 2 = 0 \implies s = -2 \] ### Conclusion: - **Negative real part**. - **System is Stable.** --- ## (e) \( \dot{x}(t) + 5x(t) = 0 \) ### Characteristic Equation: \[ s + 5 = 0 \implies s = -5 \] ### Conclusion: - **Negative real part**. - **System is Stable.** --- ## (f) \( \dfrac{d^4x(t)}{dt^4} + 18\dfrac{d^2x(t)}{dt^2} + 81x(t) = 0 \) ### Characteristic Equation: \[ s^4 + 18s^2 + 81 = 0 \] Let \( y = s^2 \): \[ y^2 + 18y + 81 = 0 \] \[ y = \frac{-18 \pm \sqrt{324 - 324}}{2} = \frac{-18}{2} = -9 \] So \( s^2 = -9 \implies s = \pm 3i \) ### Roots: - \( s = 3i, -3i, 3i, -3i \) (each root is double) ### Conclusion: - All roots are **purely imaginary and repeated**. - **System is Unstable** (since repeated imaginary roots cause unbounded oscillations). --- # **Summary Table** | Part | Roots | Stability | |------|-----------------------------------------------------|----------------------| | (a) | -2, -7 | Stable | | (b) | -0.3, -47.7 | Stable | | (c) | 1 | Unstable | | (d) | -2 | Stable | | (e) | -5 | Stable | | (f) | \(3i, -3i, 3i, -3i\) (repeated) | Unstable |