A particle of mass `m` free to move in the `x - y` plane is subjected to a force whose components are `F_(x) = - kx` and `F_(y) = - ky`, where `k` is a constant. The particle is released when `t = 0` at the point `(2, 3)`. Prove that the subsequent motion is simple harmonic along the straight line `2y - 3x = 0`.

Explanation

To solve the problem, we need to analyze the motion of a particle subjected to the given forces and prove that its motion is simple harmonic along the line defined by the equation \(2y - 3x = 0\).

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Particle:**
The forces acting on the particle are given by:
\[ F_x = -kx \quad \text{and} \quad F_y = -ky \]
where \(k\) is a constant.

2. **Determine the Resultant Force:**
The resultant force vector \(\mathbf{F}\) can be expressed as:
\[ \mathbf{F} = F_x \hat{i} + F_y \hat{j} = -kx \hat{i} - ky \hat{j} \]
This can be rewritten as:
\[ \mathbf{F} = -k(x \hat{i} + y \hat{j}) \]
This indicates that the force is proportional to the negative of the position vector \(\mathbf{r} = x \hat{i} + y \hat{j}\).

3. **Establish the Equation of Motion:**
According to Newton's second law, the motion of the particle can be described by:
\[ m \frac{d^2 \mathbf{r}}{dt^2} = \mathbf{F} \]
Substituting the expression for the force:
\[ m \frac{d^2 \mathbf{r}}{dt^2} = -k \mathbf{r} \]
This is the standard form of the equation for simple harmonic motion (SHM).

4. **Identify the Nature of Motion:**
Since the force acting on the particle is proportional to the negative of its displacement from the origin, we conclude that the particle undergoes simple harmonic motion.

5. **Determine the Line of Motion:**
The particle is released from the point \((2, 3)\). We need to check if the motion is constrained to the line defined by \(2y - 3x = 0\).

To find the slope of the line:
\[ \frac{y}{x} = \frac{3}{2} \]
This means that the motion of the particle will maintain this ratio of \(y\) to \(x\).

6. **Verify the Condition of Motion Along the Line:**
If the particle is moving along the line \(2y - 3x = 0\), we can express \(y\) in terms of \(x\):
\[ y = \frac{3}{2}x \]
This shows that as \(x\) changes, \(y\) will change accordingly, maintaining the linear relationship.

7. **Conclusion:**
Since the motion is simple harmonic and constrained to the line \(2y - 3x = 0\), we have proved that the subsequent motion of the particle is indeed simple harmonic along that line.