Find the particular solution of the differential equation.
`(dy)/(dx) = ((x sin((x)/(y))-y cos ((x)/(y)))y)/((y cos ((x)/(y))+x sin ((x)/(y)))x)`, given that y = 1 when `x = (pi)/(4)`

Explanation

`(dy)/(dx) = ((x sin((x)/(y))-y cos ((x)/(y)))y)/((y cos ((x)/(y))+x sin ((x)/(y)))x) rArr ((y cos .(x)/(y)+x sin.(x)/(y))x)/((x sin.(x)/(y)-y cos.(x)/(y))y)`
Putting x = vy
`rArr (dx)/(dy) = v + y (dv)/(dy)`
`rArr v + y (dv)/(dy) = ((y cos v + vy sin v)vy)/((vy sin v- y cos v)y)`
`y(dv)/(dy) = (v cos v + v^(2) sin v - v^(2) sin v + v cos v)/(v sin v - cos v)`
`= (2v cos v)/(v sin - cos v)`
`rArr (v sin v - cos v)/(v cos v)dv = 2(dy)/(y)`
Integrating both sides, we get
`int tan v dv - int (1)/(v)dv = 2int(dy)/(y)`
`rArr ln |sec v| - ln|v| = 2ln|y| + lnc`
`(|sec v|)/(|v|) = cy^(2)`
`rArr sec|(x)/(y)| = cy^(2)|v|`
Substitutinh `x = (pi)/(4), y = 1`, we get `c = (4sqrt(2))/(pi)`
`:. sec|(x)/(y)| = (4sqrt(2))/(pi)|(x)/(y)|.y^(2)`
or `|(x)/(y)|y^(2) cos |(x)/(y)| = (pi)/(4sqrt(2))`