The differential equations , find the particular solution satisfying the given condition:`[xsin^2(y/x)-y]dx+x dy=0; y=pi/4`when x = 1

Explanation

The given differential equation may be written as
`(dy)/(dx)=y/x-sin^(2)(y/x)`.
This is of 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)=-sinv^(2)v`
`rArr x(dv)/(dx)=-sin^(2)v`
`rArr =-"cosec"^(2)vdv=1/xdv`
`rArr int(-"cosec"^(2)v)dv=int1/xdx` [on integrating both sides]
`rArr cotv=log|x|+C`, where C is an arbitrary constant
`rArr coty/x=log|x|+C` ................(ii), `[therefore v=y/x]`.
Putting `x=1` and `y=pi/4` in (ii), we get C=1.
`therefore coty/x=log|x|+1` is the desired solution.