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

Find the particular solution, satisfying the givencondition, for the f

1.	Find the particular solution, satisfying the givencondition, for the following differential equation:`(dy)/(dx)-y/x+cos e c (y/x)=0; y=0`when `x=1`
Answer» The given differential equation may be written as `(dy)/(dx)=y/x=-"cosec"y/x`………….(i) This is the form `(dy)/(dx)=f(y/x)`. So, it is homogeneous. Putting `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get `v+x(dv)/(dx)=v-"cosec"v` `rArr -sinvdv=1/xdx` `rArr cosv=log\|x\|+C`, where C is an arbitary constant. `rArr cosy/x=log\|x\|+C...................(ii), [therefore v=y/x]`. Putting `x=1` and y=0 in (ii), we get C=1. Hence, `cosy/x=1+log\|x\|` is the required solution.