Let a dynamical system be described by the differential equation $\frac{d^2x}{dt^2} + \cos x = 0$. Which of the following differential equations describes this system in a close approximation sense for small perturbation $\Delta x$ about $x = \pi/4$?
Options:
(A) $\frac{d^2 \Delta x}{dt^2} - \frac{1}{\sqrt{2}}\Delta x = 0$
(B) $\frac{d^2 \Delta x}{dt^2} + \frac{1}{\sqrt{2}}\Delta x = 0$
(C) $\frac{d^2 \Delta x}{dt^2} + \Delta x = 0$
(D) $\frac{d^2 \Delta x}{dt^2} - \Delta x = 0$
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2007 Q.53)**
**1. Perturbation Expansion**:
Let $x(t) = x_0 + \Delta x(t)$, where $x_0 = \frac{\pi}{4}$ and $|\Delta x| \ll 1$.
Taylor expanding $\cos x$ about $x_0$:
$$\cos(x_0 + \Delta x) \approx \cos x_0 - \sin x_0 \, \Delta x = \cos(\pi/4) - \sin(\pi/4) \, \Delta x = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\Delta x$$
**2. Perturbation Dynamics**:
$$\frac{d^2(x_0 + \Delta x)}{dt^2} + \cos(x_0 + \Delta x) = \frac{d^2 \Delta x}{dt^2} + \left(\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\Delta x\right) = 0$$
The perturbed variation about the reference state gives:
$$\frac{d^2 \Delta x}{dt^2} - \frac{1}{\sqrt{2}}\Delta x = 0$$
**3. Official Master Key**:
- **Official Answer Key**: **A**