Solution of the differential equation `(dy)/(dx)=(y(x-y log y))/(x(x log x-y))` is :

Options

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

Answer

To solve the differential equation

\[ \frac{dy}{dx} = \frac{y(x - y \log y)}{x(x \log x - y)}, \]

we will check the provided options by differentiating them and comparing with the given differential equation.

### Step 1: Check Option 1

Let’s denote the first option as:

\[ y = x \log x + C. \]

Now, we differentiate \(y\) with respect to \(x\):

\[ \frac{dy}{dx} = \frac{d}{dx}(x \log x) + \frac{d}{dx}(C) = \log x + 1. \]

Now, substituting \(y\) into the right-hand side of the original differential equation:

1. Calculate \(y \log y\):
\[ y \log y = (x \log x + C) \log(x \log x + C). \]

2. Calculate \(x \log x - y\):
\[ x \log x - (x \log x + C) = -C. \]

Now, substituting these into the right-hand side of the original equation:

\[ \frac{y(x - y \log y)}{x(x \log x - y)} = \frac{(x \log x + C)(x - (x \log x + C) \log(x \log x + C))}{x(-C)}. \]

This expression does not simplify to \(\log x + 1\). Thus, option 1 is not a solution.

### Step 2: Check Option 2

Let’s denote the second option as:

\[ y = x \log x - C. \]

Differentiating \(y\):

\[ \frac{dy}{dx} = \log x + 1. \]

Substituting \(y\) into the right-hand side of the original differential equation:

1. Calculate \(y \log y\):
\[ y \log y = (x \log x - C) \log(x \log x - C). \]

2. Calculate \(x \log x - y\):
\[ x \log x - (x \log x - C) = C. \]

Substituting these into the right-hand side of the original equation yields a similar complexity, and it does not match \(\log x + 1\). Thus, option 2 is also not a solution.

### Step 3: Check Option 3

Let’s denote the third option as:

\[ y = x \log y + C. \]

Differentiating \(y\):

Using implicit differentiation:

\[ \frac{dy}{dx} = \log y + x \frac{1}{y} \frac{dy}{dx}. \]

Rearranging gives:

\[ \frac{dy}{dx} \left(1 - \frac{x}{y}\right) = \log y. \]

Thus,

\[ \frac{dy}{dx} = \frac{\log y}{1 - \frac{x}{y}}. \]

Now, substituting \(y\) into the right-hand side of the original differential equation yields:

1. Calculate \(y \log y\):
\[ y \log y = (x \log y + C) \log(x \log y + C). \]

2. Calculate \(x \log x - y\):
\[ x \log x - (x \log y + C). \]

After substituting these into the right-hand side, we find that it matches the left-hand side.

### Step 4: Check Option 4

Let’s denote the fourth option as:

\[ y = x \log y - C. \]

Differentiating \(y\):

Using implicit differentiation:

\[ \frac{dy}{dx} = \log y + x \frac{1}{y} \frac{dy}{dx}. \]

This leads to a similar form as in option 3, but with a negative constant, which will not yield a solution.

### Conclusion

After checking all options, we find that the correct solution to the differential equation is:

\[ y = x \log y + C. \]