$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}-\frac{\mathit{y}}{\mathit{x}}$ + cosec $\left(\frac{\mathit{y}}{\mathit{x}}\right)$ = 0 , y = 0 when x = 1
Let $\frac{\mathit{y}}{\mathit{x}}$ = t ⇒ y = xt
⇒ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = x $\frac{\mathit{d}\mathit{t}}{\mathit{d}\mathit{x}}$ + t
By substituting $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ in equation (i)
$\left(\mathit{x}\frac{\mathit{d}\mathit{t}}{\mathit{d}\mathit{x}}+\mathit{t}\right)$ - t + cosex t = 0
⇒ x $\frac{\mathit{d}\mathit{t}}{\mathit{d}\mathit{x}}$ = - cosec t
⇒ ∫ $\frac{\mathit{d}\mathit{t}}{\mathit{c}\mathit{o}\mathit{s}\mathit{e}\mathit{c}\mathit{t}}$ + ∫ $\frac{\mathit{d}\mathit{x}}{\mathit{x}}$ = 0
⇒ - cos t + log x = C ⇒ - cos $\left(\frac{\mathit{y}}{\mathit{x}}\right)$ + log x = C
using y 0 when x 1
- 1 + 0 = C ⇒ C = - 1
So the solution is : cos $\left(\frac{\mathit{y}}{\mathit{x}}\right)$ = log x + 1