The solution of differential equation `x sec((y)/(x))(y dx + x dy)="y cosec"((y)/(x))(x dy - y dx)` is

Options

A. `xy = "c cosec"((y)/(x))`
B. `xy^(2)"sin"(y)/(x)=c`
C. `xy "cosec"(y)/(x)=c`
D. `xy = c sin((x)/(y))`

Explanation

To solve the given differential equation:

\[ x \sec\left(\frac{y}{x}\right)(y \, dx + x \, dy) = y \csc\left(\frac{y}{x}\right)(x \, dy - y \, dx) \]

we will follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation to isolate the terms involving \(dx\) and \(dy\):

\[ x \sec\left(\frac{y}{x}\right)(y \, dx + x \, dy) - y \csc\left(\frac{y}{x}\right)(x \, dy - y \, dx) = 0 \]

### Step 2: Distribute and rearrange
Distributing the terms gives:

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

Rearranging the equation, we can group \(dx\) and \(dy\):

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

### Step 3: Factor out common terms
Now, we can factor out common terms from the equation:

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

### Step 4: Change of variables
Let \(t = \frac{y}{x}\), then \(y = tx\) and \(dy = t \, dx + x \, dt\). Substitute these into the equation:

\[ \left(tx \sec(t) + \frac{(tx)^2}{x} \csc(t)\right) dx + \left(x \sec(t) - \frac{tx}{x} \csc(t)\right) (t \, dx + x \, dt) = 0 \]

### Step 5: Simplify the equation
After substituting and simplifying, we will have:

\[ \left(t \sec(t) + t^2 \csc(t)\right) dx + \left(\sec(t) - t \csc(t)\right) (t \, dx + x \, dt) = 0 \]

### Step 6: Integrate both sides
Now, we can integrate both sides. The left-hand side can be integrated as:

\[ \int \frac{d(xy)}{xy} = \log(xy) + C_1 \]

And the right-hand side can be integrated as:

\[ \int \cot\left(\frac{y}{x}\right) d\left(\frac{y}{x}\right) = \log|\sin\left(\frac{y}{x}\right)| + C_2 \]

### Step 7: Combine results
Setting the two integrals equal gives:

\[ \log(xy) = \log|\sin\left(\frac{y}{x}\right)| + C \]

### Step 8: Exponentiate to solve for \(y\)
Exponentiating both sides results in:

\[ \frac{xy}{\sin\left(\frac{y}{x}\right)} = e^C \]

Let \(e^C = k\), where \(k\) is a constant:

\[ xy = k \sin\left(\frac{y}{x}\right) \]

### Final Solution
Thus, the solution of the differential equation is:

\[ xy = k \sin\left(\frac{y}{x}\right) \]