Find the particular solution, satisfying the given condition, for the following differential equation: `(dy)/(dx)-y/x+cos e c\ (y/x)=0; y=0` when `x=1`

Explanation

`(dy)/(dx)=(y)/(x)-(1)/(sin"(y)/(x))`
`impliesv+x(dv)/(dx)=v-(1)/(sinv)` Let `y=vx`
`implies (dy)/(dx)=v+x(dv)/(dx)`
`implies x(dv)/(dx)=-(1)/(sinv)`
`implies -sinvdv=(dx)/(x)`
`implies -intsinvdv=int(dx)/(x)+c`
`implies cosv=logx+c`
`implies cos"(y)/(x)=logx+c`
Given, `y=0` at `x=1`
`:. cos0=log1+cimpliesc=1`
Therefore, `cos"(y)/(x)=logx+1`