STATEMENT-1 : The differential equation whose general solution is `y = c_(1).x + (c_(2))/(x)` for all values of `c_(1)`, and `c_(2)` is linear equation.
and
STATEMENT-2 : The equation `y = c_(1), x + (c_(2))/(x)` has two arbitrary constants, so the corresponding differential equation is second order.

Options

A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is true, Statement-2 is true, Statement-2 is NOT a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we need to analyze the given statements and derive the corresponding differential equation from the general solution provided.

### Step-by-Step Solution:

1. **Identify the General Solution**:
The general solution given is:
\[ y = c_1 x + \frac{c_2}{x} \]
Here, \(c_1\) and \(c_2\) are arbitrary constants.

2. **Differentiate the General Solution**:
We differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = c_1 - \frac{c_2}{x^2} \]

3. **Differentiate Again**:
We differentiate \(\frac{dy}{dx}\) to find the second derivative:
\[ \frac{d^2y}{dx^2} = 0 + \frac{2c_2}{x^3} \]
This simplifies to:
\[ \frac{d^2y}{dx^2} = \frac{2c_2}{x^3} \]

4. **Express \(c_2\) in terms of \(\frac{d^2y}{dx^2}\)**:
Rearranging the equation gives:
\[ c_2 = \frac{x^3}{2} \frac{d^2y}{dx^2} \]

5. **Substitute \(c_2\) back into the first derivative**:
We can express \(c_1\) using the first derivative:
\[ c_1 = \frac{dy}{dx} + \frac{c_2}{x^2} \]
Substituting for \(c_2\):
\[ c_1 = \frac{dy}{dx} + \frac{\frac{x^3}{2} \frac{d^2y}{dx^2}}{x^2} \]
Simplifying gives:
\[ c_1 = \frac{dy}{dx} + \frac{x}{2} \frac{d^2y}{dx^2} \]

6. **Substituting \(c_1\) and \(c_2\) back into the original equation**:
Substitute \(c_1\) and \(c_2\) back into the general solution:
\[ y = \left(\frac{dy}{dx} + \frac{x}{2} \frac{d^2y}{dx^2}\right)x + \frac{\frac{x^3}{2} \frac{d^2y}{dx^2}}{x} \]
This simplifies to:
\[ y = x \frac{dy}{dx} + \frac{x^2}{2} \frac{d^2y}{dx^2} + \frac{x^2}{2} \frac{d^2y}{dx^2} \]
Combining terms results in:
\[ y = x \frac{dy}{dx} + x^2 \frac{d^2y}{dx^2} \]

7. **Final Form of the Differential Equation**:
The resulting differential equation is:
\[ y - x \frac{dy}{dx} - x^2 \frac{d^2y}{dx^2} = 0 \]

8. **Determine the Order of the Differential Equation**:
The highest derivative present is \(\frac{d^2y}{dx^2}\), indicating that this is a second-order differential equation.

### Conclusion:
- **Statement 1** is true: The differential equation whose general solution is \(y = c_1 x + \frac{c_2}{x}\) is indeed a linear equation.
- **Statement 2** is also true: The equation has two arbitrary constants, thus the corresponding differential equation is of second order.