# Find the particular solution, satisfying the given condition, for the following differential equation : $\large \frac{dy}{dx}-\frac{y}{x}$$+cosec \bigg( \large\frac{y}{x} \bigg)$$=0; y=0\: when \: x = 1$

Toolbox:
• To solve homogeneous differential equation, put $y = vx$ and $\large\frac{dy}{dx}$$= v+x\large\frac{dv}{dx} Step 1: Let us rearrange and write the equation as \large\frac{dy}{dx} = (\large\frac{y}{x})$$ - cosec(\large\frac{y}{x})$
Using the information in the tool box,
$v+x\large\frac{dv}{dx}$$= v + cosec v cancelling v on both sides we get, x\large\frac{dv}{dx }$$= - cosec v$
Seperating the variables we get,
$\large\frac{dv}{cosec v }= \frac{- dx}{x}$
Since $\large\frac{1}{cosec v}$$= \sin v \sin v dv = -\large\frac{ dx}{x} Step 2: Integrating on both sides we get, \int \sin vdv = - \int\large\frac{dx}{x} - \cos v = -\log x - C \cos v = \log x + C Substituting for v we get, \cos(\large\frac{y}{x})$$= \log x + log c$
Step 3:
Given $y = 0$ and $x = 1$ we get,
$\cos(0) = \log(1) + \log C$, since $\cos 0 = 1$ and $\log 1 = 0$
$\log_e C= 1$
$C = e^1$
$C=e$
Substituting for $C$ we get