`[x sin^(2)((y)/(x)) - y] dx + x dy = 0, y = (pi)/(4)` when x = 1

Answer

`cot(y/x)=log_(e)|ex|`

Explanation

`[xsin^(2)(y/x)-y]dx+xdy=0`
or `(dy)/(dx)=v+(xdv)/(dx)`
Therefore, given equation reduces to
`v+x(dv)/(dx)=v-sin^(2)v`
or `x(dv)/(dx) = -sin^(2)v`
or `"cosec "^(2)vdv=-(dx)/(x)`
Integrating both sides, we get
`-cotv=-log|x|-logC`
or `cot(y/x)=log|Cx|`...............(2)
Now, `y=pi/4` at `x=1`.
`therefore cot(pi/4) = log|C|`
or `1=log C`
or C=e
Substituting C=e in equation (2), we get
`cot(y/x)=log|ex|`
This is the required solution of the given differential equation