$\frac{dy}{dx} - \frac{y}{x}$ + cosec $\left(\frac{y}{x}\right)$ = 0 , y = 0 when x = 1 Let $\frac{y}{x}$ = t ⇒ y = xt ⇒ $\frac{dy}{dx}$ = x $\frac{dt}{dx}$ + t By substituting $\frac{dy}{dx}$ in equation (i) $\left(x\frac{dt}{dx}+t\right)$ - t + cosex t = 0 ⇒ x $\frac{dt}{dx}$ = - cosec t ⇒ ∫ $\frac{dt}{\text{cosect}}$ + ∫ $\frac{dx}{x}$ = 0 ⇒ - cos t + log x = C ⇒ - cos $\left(\frac{y}{x}\right)$ + log x = C using y 0 when x 1 - 1 + 0 = C ⇒ C = - 1 So the solution is : cos $\left(\frac{y}{x}\right)$ = log x + 1