dy

dx

−

y

x

+ cosec (

y

x

) = 0 , y = 0 when x = 1
Let

y

x

= t ⇒ y = xt
⇒

dy

dx

= x

dt

dx

+ t
By substituting

dy

dx

in equation (i)
(x

dt

dx

+t) - t + cosex t = 0
⇒ x

dt

dx

= - cosec t
⇒ ∫

dt

cosect

+ ∫

dx

x

= 0
⇒ - cos t + log x = C ⇒ - cos (

y

x

) + log x = C
using y 0 when x 1
- 1 + 0 = C ⇒ C = - 1
So the solution is : cos (

y

x

) = log x + 1