Find the particular solution of the differential equation given by $\displaystyle x^{2} \frac{d y}{d x}-x y=x^{2} \cos ^{2}\left(\frac{y}{2 x}\right)$, given that when $\displaystyle x=1, y=\frac{\pi}{2}$.

$$\frac{dy}{dx} = \frac{y}{x} + \cos^2\left(\frac{y}{2x}\right) Put $\displaystyle y = vx$ so that $\displaystyle \dfrac{dy}{dx} = v + x\dfrac{dv}{dx}$ \Rightarrow v + x\frac{dv}{dx} = v + \cos^2\left(\frac{v}{2}\right) \Rightarrow \int \sec^2\left(\frac{v}{2}\right) dv = \int \frac{1}{x}\, dx \Rightarrow 2\tan\left(\frac{v}{2}\right) = \log|x| + C \Rightarrow 2\tan\left(\frac{y}{2x}\right) = \log|x| + C 2\tan\frac{\pi}{4} = \log 1 + C \Rightarrow C = 2 \Rightarrow 2\tan\left(\frac{y}{2x}\right) = \log|x| + 2$$