`(dy)/(dx) = (xy+y)/(xy+x)`, then the solution of differential equation is

Options

A. `y = xe^(x) + c`
B. `y = e^(x) + c`
C. `y = c x e^(x-y)`
D. y = x+c

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 can rewrite the right-hand side of the equation:
\[ \frac{dy}{dx} = \frac{y(x + 1)}{x(y + 1)} \]

### Step 2: Separate the variables
We can separate the variables \(y\) and \(x\):
\[ \frac{1 + \frac{1}{y}}{1 + \frac{1}{x}} dy = dx \]

### Step 3: Integrate both sides
Now we will integrate both sides:
\[ \int \left(1 + \frac{1}{y}\right) dy = \int \left(1 + \frac{1}{x}\right) dx \]

The left-hand side integrates to:
\[ y + \ln |y| \]
And the right-hand side integrates to:
\[ x + \ln |x| + C \]

### Step 4: Combine the results
Equating both sides gives us:
\[ y + \ln |y| = x + \ln |x| + C \]

### Step 5: Solve for \(y\)
This equation can be rewritten in a more manageable form:
\[ y - x = \ln |x| - \ln |y| + C \]

### Step 6: Exponentiate to solve for \(y\)
We can express this in exponential form:
\[ y = x e^{C} e^{\ln |x| - \ln |y|} = k \frac{x}{y} \]
where \(k = e^{C}\).

### Final Solution
Thus, we can express the solution in a more compact form:
\[ y = x e^{x - y + C} \]