Solve the initial value problem: `x\ (dy)/(dx)-y+xsin(y/x)=0,\ y(2)=x`

Explanation

`x \frac{d y}{d x}-y+x \sin (\frac{y}{x})=0, y(2)=x`

It is homogeneous equation,

Put `y=v x`

And `x \frac{d v}{d x}+v=\frac{d y}{d x}`

So,

`x \frac{d v}{d x}+v=-\sin (\frac{v x}{x})+\frac{v x}{x} `

`x \frac{d v}{d x}=-\sin v `

`frac{d v}{\sin v}=-\frac{d x}{x} `

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