The solution of `xsin((y)/(x))dy={ysin((y)/(x))-x}dx,` is given by

Options

A. `logx+cos((y)/(x))=logC`
B. `logx-cos((y)/(x))=logC`
C. `log((x)/(y))-cos ((y)/(x))=logC`
D. None of these

Answer

To solve the differential equation \( x \sin\left(\frac{y}{x}\right) dy = \left(y \sin\left(\frac{y}{x}\right) - x\right) dx \), we can follow these steps:

### Step 1: Rewrite the Equation
First, we rewrite the given equation in a more manageable form:
\[ x \sin\left(\frac{y}{x}\right) dy - \left(y \sin\left(\frac{y}{x}\right) - x\right) dx = 0 \]

### Step 2: Identify Homogeneous Form
This equation is a homogeneous differential equation. We can use the substitution \( v = \frac{y}{x} \), which implies \( y = vx \).

### Step 3: Differentiate
Differentiating \( y = vx \) with respect to \( x \) gives:
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]

### Step 4: Substitute into the Equation
Substituting \( y = vx \) and \( \frac{dy}{dx} \) into the original equation:
\[ x \sin(v) \left(v + x \frac{dv}{dx}\right) = \left(vx \sin(v) - x\right) \]

### Step 5: Simplify
This simplifies to:
\[ x \sin(v) v + x^2 \sin(v) \frac{dv}{dx} = vx \sin(v) - x \]
Rearranging gives:
\[ x^2 \sin(v) \frac{dv}{dx} = -x \]

### Step 6: Separate Variables
Now, we separate variables:
\[ \sin(v) \frac{dv}{dx} = -\frac{1}{x} \]

### Step 7: Integrate Both Sides
Integrating both sides:
\[ \int \sin(v) dv = -\int \frac{1}{x} dx \]
This results in:
\[ -\cos(v) = -\ln|x| + C \]

### Step 8: Substitute Back
Substituting back \( v = \frac{y}{x} \):
\[ -\cos\left(\frac{y}{x}\right) = -\ln|x| + C \]
Rearranging gives:
\[ \ln|x| - \cos\left(\frac{y}{x}\right) = C \]

### Final Solution
Thus, the solution of the differential equation is:
\[ \ln|x| - \cos\left(\frac{y}{x}\right) = C \]