The solution of the `( dy )/(dx) = (xy+y)/(xy+x)`is :

Options

A. `(1)/( x+y) = x^(2) + 1 + Ce ^(x)`
B. `(1) /(( x+y)^2)=x^2+1 + Ce^(x^2)`
C. `(1)/((x+y)^2) = X+1+ Ce ^(x)`
D. none of these

Answer

To solve the differential equation \(\frac{dy}{dx} = \frac{xy + y}{xy + x}\), we will follow these steps:

### Step 1: Simplify the Equation
We start with the given equation:
\[ \frac{dy}{dx} = \frac{xy + y}{xy + x} \]
We can factor out \(y\) from the numerator and \(x\) from the denominator:
\[ \frac{dy}{dx} = \frac{y(x + 1)}{x(y + 1)} \]

### Step 2: Cross-Multiply
Now, we cross-multiply to separate the variables:
\[ (y + 1) dy = \frac{(x + 1)}{x} dx \]

### Step 3: Integrate Both Sides
Next, we integrate both sides:
\[ \int (y + 1) dy = \int \frac{(x + 1)}{x} dx \]
The left side becomes:
\[ \int (y + 1) dy = \frac{y^2}{2} + y \]
The right side can be simplified:
\[ \int \left(1 + \frac{1}{x}\right) dx = x + \log |x| \]

### Step 4: Combine the Results
Combining the results from both integrals gives us:
\[ \frac{y^2}{2} + y = x + \log |x| + C \]

### Step 5: Rearrange the Equation
We can rearrange the equation:
\[ \frac{y^2}{2} + y - x - \log |x| - C = 0 \]

### Final Solution
Thus, the implicit solution of the differential equation is:
\[ \frac{y^2}{2} + y - x - \log |x| = C \]