Which one of the following differential equations has a periodic solution ?

Options

A. `(d^(2) x)/(dt^(2)) + mu x = 0`
B. `(d^(2) x)/(dt^(2)) - mu x = 0`
C. `x "" (dx)/(dt) + mu t = 0`
D. `(dx)/(dt) + mu x t = 0`

Answer

To determine which differential equation has a periodic solution, we can analyze the standard form of differential equations known to produce periodic solutions.

### Step-by-Step Solution:

1. **Understand the Concept of Periodic Solutions**:
- A periodic solution is a function that repeats its values at regular intervals, typically described by sine or cosine functions. The simplest example of such a solution arises from the equations of Simple Harmonic Motion (SHM).

2. **Identify the Standard Form of SHM**:
- The standard differential equation for SHM is given by:
\[ \frac{d^2x}{dt^2} + \mu x = 0 \]
- Here, \( \mu \) is a constant that relates to the stiffness of the system.

3. **Solve the SHM Equation**:
- The general solution to this differential equation can be expressed as:
\[ x(t) = A \cos(\omega t - \phi) \quad \text{or} \quad x(t) = A \sin(\omega t - \phi) \]
- Where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

4. **Recognize the Periodicity**:
- Both forms of the solution (sine and cosine) are periodic functions with a period of \( T = \frac{2\pi}{\omega} \). This means that the function will repeat its values every \( T \) seconds.

5. **Evaluate Given Options**:
- If the question provides multiple differential equations, we would check each one to see if it can be transformed into the form of the SHM equation. The one that can be expressed in this form will have a periodic solution.

6. **Conclusion**:
- Based on the knowledge of SHM, the answer to the question is the differential equation that resembles the form \( \frac{d^2x}{dt^2} + \mu x = 0 \).

### Final Answer:
- The differential equation that has a periodic solution is option **A**.