Solve the `(xdy-ydx)ysin(y/x)=(ydx+xdy)xcos(y/x)`

Answer

To solve the equation \((xdy - ydx) \sin\left(\frac{y}{x}\right) = (ydx + xdy) x \cos\left(\frac{y}{x}\right)\), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the given equation for clarity:
\[ (xdy - ydx) \sin\left(\frac{y}{x}\right) = (ydx + xdy) x \cos\left(\frac{y}{x}\right) \]

### Step 2: Expand both sides
Distributing terms on both sides gives:
\[ xydy \sin\left(\frac{y}{x}\right) - y^2dx \sin\left(\frac{y}{x}\right) = ydx \cdot x \cos\left(\frac{y}{x}\right) + x^2dy \cos\left(\frac{y}{x}\right) \]

### Step 3: Divide by \(dx\)
Dividing the entire equation by \(dx\) (assuming \(dx \neq 0\)):
\[ \frac{d y}{d x} \cdot xy \sin\left(\frac{y}{x}\right) - y^2 \sin\left(\frac{y}{x}\right) = y \cdot x \cos\left(\frac{y}{x}\right) + x^2 \frac{d y}{d x} \cos\left(\frac{y}{x}\right) \]

### Step 4: Collect terms involving \(\frac{dy}{dx}\)
Rearranging gives:
\[ \frac{d y}{d x} \cdot (xy \sin\left(\frac{y}{x}\right) - x^2 \cos\left(\frac{y}{x}\right)) = y \cdot x \cos\left(\frac{y}{x}\right) + y^2 \sin\left(\frac{y}{x}\right) \]

### Step 5: Isolate \(\frac{dy}{dx}\)
Now isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{y \cdot x \cos\left(\frac{y}{x}\right) + y^2 \sin\left(\frac{y}{x}\right)}{xy \sin\left(\frac{y}{x}\right) - x^2 \cos\left(\frac{y}{x}\right)} \]

### Step 6: Substitute \(v = \frac{y}{x}\)
Let \(v = \frac{y}{x}\) which implies \(y = vx\). Then, differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]

### Step 7: Substitute into the equation
Substituting \(y\) and \(\frac{dy}{dx}\) into the equation gives:
\[ v + x \frac{dv}{dx} = \frac{(vx) \cdot x \cos(v) + (vx)^2 \sin(v)}{(vx) \cdot x \sin(v) - x^2 \cos(v)} \]

### Step 8: Simplify and solve for \(\frac{dv}{dx}\)
After substituting and simplifying, we can isolate \(\frac{dv}{dx}\) and solve the resulting differential equation.

### Step 9: Integrate both sides
Integrate both sides to find \(v\) in terms of \(x\).

### Step 10: Back-substitute to find \(y\)
Finally, back-substitute \(v = \frac{y}{x}\) to express \(y\) in terms of \(x\).