Problem Statement
Find the particular solution of the differential equation $\frac{dy}{dx} - \frac{y}{x} + \csc\left(\frac{y}{x}\right) = 0$, given that $y = 0$ when $x = 1$.
Verified Solution & Marking Scheme
Identify Homogeneous Equation
$\frac{dy}{dx} = \frac{y}{x} - \csc\left(\frac{y}{x}\right)$
Substitute y = vx
Let $y = vx \implies \frac{dy}{dx} = v + x\frac{dv}{dx}$:
$v + x\frac{dv}{dx} = v - \csc v \implies x\frac{dv}{dx} = -\csc v = -\frac{1}{\sin v}$
$\sin v \, dv = -\frac{dx}{x}$
Integrate Both Sides
$\int \sin v \, dv = -\int \frac{dx}{x} \implies -\cos v = -\ln|x| + C \implies \cos\left(\frac{y}{x}\right) = \ln|x| + C'$
Apply Initial Condition y(1) = 0
$\cos(0) = \ln(1) + C' \implies 1 = 0 + C' \implies C' = 1$
Particular solution: $\cos\left(\frac{y}{x}\right) = \ln|x| + 1$.