Show that the differential equation `x(dy)/(dx)sin(y/x)+x-ysin(y/x)=0` is homogenous. Find the particular solution of this differential equation, given that `x=1` when `y=pi/2`.

Explanation

To show that the differential equation \( x \frac{dy}{dx} \sin\left(\frac{y}{x}\right) + x - y \sin\left(\frac{y}{x}\right) = 0 \) is homogeneous and to find the particular solution given \( x = 1 \) when \( y = \frac{\pi}{2} \), we will follow these steps:

### Step 1: Verify Homogeneity
We start with the given differential equation:

\[ x \frac{dy}{dx} \sin\left(\frac{y}{x}\right) + x - y \sin\left(\frac{y}{x}\right) = 0. \]
...