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

### Step 2: Simplify both sides
We can express \(\sec\) and \(\csc\) in terms of sine and cosine:
\[ \sec\left(\frac{y}{x}\right) = \frac{1}{\cos\left(\frac{y}{x}\right)}, \quad \csc\left(\frac{y}{x}\right) = \frac{1}{\sin\left(\frac{y}{x}\right)} \]
Substituting these into the equation gives:
\[ x \frac{1}{\cos\left(\frac{y}{x}\right)}(y \, dx + x \, dy) = y \frac{1}{\sin\left(\frac{y}{x}\right)}(x \, dy - y \, dx) \]

### Step 3: Multiply through by \(\sin\left(\frac{y}{x}\right) \cos\left(\frac{y}{x}\right)\)
To eliminate the trigonometric functions, we multiply both sides by \(\sin\left(\frac{y}{x}\right) \cos\left(\frac{y}{x}\right)\):
\[ x \sin\left(\frac{y}{x}\right)(y \, dx + x \, dy) = y \cos\left(\frac{y}{x}\right)(x \, dy - y \, dx) \]

### Step 4: Rearranging terms
Rearranging the terms leads to:
\[ x y \sin\left(\frac{y}{x}\right) \, dx + x^2 \sin\left(\frac{y}{x}\right) \, dy = y x \cos\left(\frac{y}{x}\right) \, dy - y^2 \cos\left(\frac{y}{x}\right) \, dx \]

### Step 5: Grouping terms
Group all \(dx\) and \(dy\) terms:
\[ (x y \sin\left(\frac{y}{x}\right) + y^2 \cos\left(\frac{y}{x}\right)) \, dx = (y x \cos\left(\frac{y}{x}\right) - x^2 \sin\left(\frac{y}{x}\right)) \, dy \]

### Step 6: Dividing by \(x^2 y\)
Dividing both sides by \(x^2 y\):
\[ \frac{(x y \sin\left(\frac{y}{x}\right) + y^2 \cos\left(\frac{y}{x}\right))}{x^2 y} \, dx = \frac{(y x \cos\left(\frac{y}{x}\right) - x^2 \sin\left(\frac{y}{x}\right))}{x^2 y} \, dy \]

### Step 7: Integrating both sides
Now, we can integrate both sides:
\[ \int \frac{d(xy)}{xy} = \int \cot\left(\frac{y}{x}\right) \, d\left(\frac{y}{x}\right) \]

### Step 8: Solving the integrals
The left side integrates to:
\[ \log(xy) \]
The right side integrates to:
\[ \log(\sin\left(\frac{y}{x}\right)) \]

### Step 9: Final equation
Setting the two integrals equal gives:
\[ \log(xy) = \log(\sin\left(\frac{y}{x}\right)) + C \]
Exponentiating both sides results in:
\[ xy = C \sin\left(\frac{y}{x}\right) \]

### Final Answer
Thus, the solution of the differential equation is:
\[ xy = C \sin\left(\frac{y}{x}\right) \]