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

Given that , `sin^(2)((y)/(x))-(y)/(x)+(dy)/(dx)=0`
`implies (dy)/(dx)=(y)/(x)-sin^(2)((y)/(x))`……….`(1)`
Given differential equation is homogenous.
Let `y=vximplies(dy)/(dx)=v+x(dv)/(dx)`
From equation `(1)`, `v+x(dv)/(dx)=v-sin^(2)v`
`x(dv)/(dx)=-sin^(2)v`
`implies cosec^(2)vdv=-(1)/(x)dx`
On integration
`intcosec^(2)vdv=-int(dx)/(x)implies-cotv=-log|x|+C`
`implies log|x|-cotv=Cimplies log|x|-cot((y)/(x))=C`.........`(2)`
when `x=1`, then `y=(pi)/(4)implieslog|1|-cot(pi)/(4)=C`
`implies C=0-1=-1`
put the value of `C` in equation `(2)`,
`log|x|-cot((y)/(x))=-1`
`implies log|x|-cot((y)/(x)=-loge`
`implies cot((y)/(x))=log|ex|`
which is the required solution of the given equation.