STATEMENT-1 : Solution of the differential equation `xdy-y dx = y dy` is `ye^(x//y) = c`.
and
STATEMENT-2 : Given differential equation can be re-written as `d((x)/(y)) = -(dy)/(y)`.

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 given differential equation \( x \, dy - y \, dx = y \, dy \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ x \, dy - y \, dx = y \, dy \]
Rearranging this, we can write:
\[ x \, dy - y \, dy = y \, dx \]
This simplifies to:
\[ (x - y) \, dy = y \, dx \]

### Step 2: Dividing by \( y \)
Next, we divide both sides by \( y \) (assuming \( y \neq 0 \)):
\[ \frac{x - y}{y} \, dy = dx \]
This can be rewritten as:
\[ \left(\frac{x}{y} - 1\right) \, dy = dx \]

### Step 3: Expressing in Differential Form
Now, we can express this in differential form:
\[ d\left(\frac{x}{y}\right) = -\frac{dy}{y} \]

### Step 4: Integrating Both Sides
We will integrate both sides:
\[ \int d\left(\frac{x}{y}\right) = \int -\frac{dy}{y} \]
The left side integrates to:
\[ \frac{x}{y} = -\ln|y| + C \]
where \( C \) is the constant of integration.

### Step 5: Rearranging the Result
Rearranging this gives:
\[ \ln|y| + \frac{x}{y} = C \]
Exponentiating both sides results in:
\[ y \cdot e^{\frac{x}{y}} = e^C \]
Letting \( e^C = k \) (a constant), we have:
\[ y \cdot e^{\frac{x}{y}} = k \]
Thus, we can write:
\[ y \cdot e^{\frac{x}{y}} = c \]
where \( c \) is a constant.

### Conclusion
The solution of the differential equation \( x \, dy - y \, dx = y \, dy \) is indeed:
\[ y e^{\frac{x}{y}} = c \]