Show that the differential equation: `(xcos(y/x))(ydx+xdy)=(ysin(y/x))(xdy-ydx)` is homogenous and solve it.

Explanation

To solve the given differential equation \((x \cos(y/x))(y \, dx + x \, dy) = (y \sin(y/x))(x \, dy - y \, dx)\), we will follow the steps below:

### Step 1: Show that the equation is homogeneous.

To prove that the equation is homogeneous, we will replace \(x\) with \(\lambda x\) and \(y\) with \(\lambda y\) and check if the equation remains unchanged after simplification.

Substituting \(x = \lambda x\) and \(y = \lambda y\):

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