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

Options

A. `x/y+log (xy)=C`
B. `y/x +log (xy)=C`
C. ` x/y + y log x = C`
D. ` x/y + x log y = C`

Explanation

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

### Step 1: Rearranging the Equation
First, we can rearrange the given equation to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{y(x+y)}{x(x-y)} \]

### Step 2: Substituting \( y = vx \)
Next, we will use the substitution \( y = vx \), where \( v \) is a function of \( x \). Then, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]

### Step 3: Substitute \( y \) and \( \frac{dy}{dx} \)
Substituting \( y = vx \) and \(\frac{dy}{dx}\) into the rearranged equation gives:
\[ v + x \frac{dv}{dx} = \frac{vx(x + vx)}{x(x - vx)} \]
This simplifies to:
\[ v + x \frac{dv}{dx} = \frac{v(x^2 + vx)}{x(x - vx)} = \frac{v(x + v)}{(1 - v)} \]

### Step 4: Simplifying the Equation
Now, we can simplify the equation further:
\[ x \frac{dv}{dx} = \frac{v(x + v)}{(1 - v)} - v \]
\[ x \frac{dv}{dx} = \frac{v(x + v - v + v^2)}{(1 - v)} = \frac{v(x + v^2)}{(1 - v)} \]

### Step 5: Separating Variables
Now, we can separate the variables:
\[ \frac{(1 - v)}{v(x + v^2)} dv = \frac{dx}{x} \]

### Step 6: Integrating Both Sides
Integrate both sides:
\[ \int \frac{(1 - v)}{v(x + v^2)} dv = \int \frac{dx}{x} \]
This can be split into two integrals:
\[ \int \left( \frac{1}{v} - \frac{1}{x + v^2} \right) dv = \log |x| + C \]

### Step 7: Solving the Integrals
The left side can be integrated:
\[ \log |v| - \frac{1}{2} \log |x + v^2| = \log |x| + C \]

### Step 8: Back Substituting for \( v \)
Recall that \( v = \frac{y}{x} \). Substitute back:
\[ \log \left|\frac{y}{x}\right| - \frac{1}{2} \log |x + \left(\frac{y}{x}\right)^2| = \log |x| + C \]

### Step 9: Final Rearrangement
Rearranging gives us the implicit solution of the differential equation:
\[ \frac{y}{x} - \frac{1}{2} \log |x + \frac{y^2}{x^2}| = \log |x| + C \]

### Final Solution
Thus, the solution of the differential equation is:
\[ y = x \left( \text{some function of } x \right) \]