The solution of the differential equation `{1/x-y^(2)/(x-y)^(2)}dx+{x^(2)/(x-y)^(2)-1/y}dy=0` is

Options

A. `"ln"|x/y|+(xy)/(x-y)=c`
B. `(xy)/(x-y)=ce^(x//y)`
C. `"ln"|xy|+(x^(4)y^(4))/4=C`
D. None of these

Explanation

To solve the given differential equation

\[ \frac{1}{x - \frac{y^2}{(x - y)^2}} dx + \left(\frac{x^2}{(x - y)^2} - \frac{1}{y}\right) dy = 0, \]

we will follow these steps:

### Step 1: Rewrite the differential equation
We start by rewriting the equation in a more manageable form:

\[ \left( \frac{1}{x} - \frac{y^2}{(x - y)^2} \right) dx + \left( \frac{x^2}{(x - y)^2} - \frac{1}{y} \right) dy = 0. \]

### Step 2: Combine the terms
Next, we can combine the terms in the equation. We will take the common denominator for the terms involving \(dx\) and \(dy\):

\[ \frac{(x - y)^2}{(x - y)^2} \left( \frac{1}{x} - \frac{y^2}{(x - y)^2} \right) dx + \frac{(x - y)^2}{(x - y)^2} \left( \frac{x^2}{(x - y)^2} - \frac{1}{y} \right) dy = 0. \]

### Step 3: Simplify the equation
Now, we simplify the equation further:

\[ \frac{(x - y)^2}{x} dx - \frac{y^2}{(x - y)^2} dx + \left( \frac{x^2}{(x - y)^2} - \frac{1}{y} \right) dy = 0. \]

### Step 4: Separate variables
We can rearrange the equation to separate the variables:

\[ \frac{dx}{x} - \frac{dy}{y} + \left( \frac{dy}{y^2} - \frac{dx}{x^2} \right) = 0. \]

### Step 5: Integrate both sides
Now we integrate both sides:

1. Integrate \(\frac{dx}{x}\) to get \(\ln |x|\).
2. Integrate \(-\frac{dy}{y}\) to get \(-\ln |y|\).
3. Integrate \(\frac{dy}{y^2}\) to get \(-\frac{1}{y}\).
4. Integrate \(-\frac{dx}{x^2}\) to get \(\frac{1}{x}\).

### Step 6: Combine the results
Combining these results, we have:

\[ \ln |x| - \ln |y| + \frac{1}{y} - \frac{1}{x} = C, \]

where \(C\) is a constant.

### Step 7: Exponentiate to solve for the function
Exponentiating both sides gives us:

\[ \frac{x}{y} = e^{C} \cdot \left( \frac{1}{y} - \frac{1}{x} \right). \]

### Final Step: Rearranging to find the solution
Rearranging gives us the final solution in implicit form:

\[ \ln \left( \frac{x}{y} \right) + \frac{xy}{x - y} = C. \]

Thus, the solution of the differential equation is:

\[ \ln \left( \frac{x}{y} \right) + \frac{xy}{x - y} = C. \]