If the solution of the differential equation `(1+e^((x)/(y)))dx+e^((x)/(y))(1-(x)/(y))dy=0` is `x+kye^((x)/(y))=C` (where, C is an arbitrary constant), then the value of k is equal to

Answer

To solve the given differential equation \((1 + e^{\frac{x}{y}})dx + e^{\frac{x}{y}}\left(1 - \frac{x}{y}\right)dy = 0\) and find the value of \(k\) in the solution \(x + kye^{\frac{x}{y}} = C\), we can follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the equation:
\[ (1 + e^{\frac{x}{y}})dx + e^{\frac{x}{y}}\left(1 - \frac{x}{y}\right)dy = 0 \]

### Step 2: Rearranging the Terms
Rearranging the terms gives:
\[ (1 + e^{\frac{x}{y}})dx = -e^{\frac{x}{y}}\left(1 - \frac{x}{y}\right)dy \]

### Step 3: Factor Out \(e^{\frac{x}{y}}\)
We can factor out \(e^{\frac{x}{y}}\) from the second term:
\[ (1 + e^{\frac{x}{y}})dx + e^{\frac{x}{y}}\left(-1 + \frac{x}{y}\right)dy = 0 \]

### Step 4: Combine Terms
Now we can combine the terms:
\[ dx + e^{\frac{x}{y}}dy - \frac{x}{y}e^{\frac{x}{y}}dy = 0 \]
This simplifies to:
\[ dx + e^{\frac{x}{y}}dy - \frac{x}{y}e^{\frac{x}{y}}dy = 0 \]

### Step 5: Simplifying Further
We can rewrite the equation as:
\[ dx + e^{\frac{x}{y}}\left(1 - \frac{x}{y}\right)dy = 0 \]

### Step 6: Change of Variables
Let \(z = \frac{x}{y}\), then \(x = zy\) and \(dx = ydz + zdy\). Substitute these into the equation:
\[ (y dz + z dy) + e^{z}(1 - z)dy = 0 \]

### Step 7: Collecting Like Terms
This gives:
\[ y dz + (z + e^{z}(1 - z))dy = 0 \]

### Step 8: Separating Variables
Now, we separate the variables:
\[ \frac{dz}{dy} = -\frac{z + e^{z}(1 - z)}{y} \]

### Step 9: Integrating
Integrate both sides:
\[ \int dz = -\int \frac{z + e^{z}(1 - z)}{y} dy \]

### Step 10: Solve the Integral
After integrating, we will have:
\[ z + e^{z} = C \]

### Step 11: Substitute Back
Substituting back \(z = \frac{x}{y}\):
\[ \frac{x}{y} + e^{\frac{x}{y}} = C \]

### Step 12: Rearranging to Find \(k\)
Rearranging gives:
\[ x + kye^{\frac{x}{y}} = C \]
Comparing with the required form, we find that \(k = 1\).

### Conclusion
Thus, the value of \(k\) is:
\[ \boxed{1} \]