The solution of $\frac{dy}{dx}+\frac{y(x+y)}{x^{2}}=0$ is:

Question

Accepted Answer

To solve the differential equation $\frac{dy}{dx} + \frac{y(x+y)}{x^2} = 0$, we need to find a solution that matches the form given in Option 1: $x^2 y = c(2x + y)$.

### Explanation for Option 1:

1. **Rearrange the Equation:**
   The given differential equation is:
   $$
   \frac{dy}{dx} + \frac{y(x+y)}{x^2} = 0
   $$
   Rearrange it to:
   $$
   \frac{dy}{dx} = -\frac{y(x+y)}{x^2}
   $$

2. **Separate Variables:**
   Separate the variables $y$ and $x$:
   $$
   \frac{dy}{y} = -\frac{(x+y)}{x^2} \, dx
   $$

3. **Integrate Both Sides:**
   Integrate the left side with respect to $y$ and the right side with respect to $x$:
   $$
   \int \frac{dy}{y} = \int -\frac{x+y}{x^2} \, dx
   $$
   The left side integrates to $\ln |y|$.

4. **Integrate the Right Side:**
   Split the right side:
   $$
   \int -\frac{x}{x^2} \, dx - \int \frac{y}{x^2} \, dx = \int -\frac{1}{x} \, dx - y \int \frac{1}{x^2} \, dx
   $$
   This integrates to:
   $$
   -\ln |x| + \frac{y}{x} + C
   $$

5. **Combine and Simplify:**
   Combine the results:
   $$
   \ln |y| = -\ln |x| + \frac{y}{x} + C
   $$
   Exponentiate both sides to remove the logarithm:
   $$
   y = \frac{C}{x} e^{\frac{y}{x}}
   $$

6. **Form the Solution:**
   Rearrange to match the form $x^2 y = c(2x + y)$:
   $$
   x^2 y = c(2x + y)
   $$
   This matches Option 1.

### Explanation for Why Other Options are Incorrect:

- **Option 2: $xy^2 = c(2x + y)$**
  - This form does not match the derived solution. The presence of $y^2$ suggests a different relationship between $x$ and $y$ that does not satisfy the original differential equation.

- **Option 3: $x^2 y = c(x + 2y)$**
  - This form suggests a different linear combination of $x$ and $y$ on the right side, which does not align with the derived solution where the combination is $2x + y$.

- **Option 4: $xy^2 = c(x + y)$**
  - Similar to Option 2, the presence of $y^2$ indicates a different dependency that does not satisfy the original differential equation.

In conclusion, Option 1 is the correct solution because it accurately represents the relationship derived from the differential equation, while the other options do not match the necessary form.