A particle is moving along X-axis Its acceleration at time t is proportional to its velocity at that time. The differential equation of the motion of the particle is

Options

A. `(d^(2)x)/(dt^(2))-k(dx)/(dt)=0,kgt0`
B. `(d^(2)x)/(dt^(2))+k(dx)/(dt)=0,kgt0`
C. `(d^(2)x)/(dt^(2))-k=0,kgt0`
D. `(d^(2)x)/(dt^(2))+k=0,kgt0`

Answer

To solve the problem, we need to derive the differential equation of motion for a particle whose acceleration is proportional to its velocity. Let's go through the steps systematically.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( x(t) \) be the position of the particle at time \( t \).
- Let \( v(t) = \frac{dx}{dt} \) be the velocity of the particle.
- Let \( a(t) = \frac{d^2x}{dt^2} \) be the acceleration of the particle.

2. **Establish the Relationship**:
- According to the problem, the acceleration \( a \) is proportional to the velocity \( v \). This can be expressed mathematically as:
\[ a = k v \]
where \( k \) is a positive constant of proportionality (i.e., \( k > 0 \)).

3. **Substitute the Definitions**:
- Substitute the definitions of acceleration and velocity into the equation:
\[ \frac{d^2x}{dt^2} = k \frac{dx}{dt} \]

4. **Rearrange the Equation**:
- Rearranging the equation gives us:
\[ \frac{d^2x}{dt^2} - k \frac{dx}{dt} = 0 \]

5. **Final Form of the Differential Equation**:
- The resulting differential equation that describes the motion of the particle is:
\[ \frac{d^2x}{dt^2} - k \frac{dx}{dt} = 0 \]

### Conclusion:
Thus, the differential equation of the motion of the particle is:
\[ \frac{d^2x}{dt^2} - k \frac{dx}{dt} = 0 \]