2020 EJC P1 Q9

Given \(x\) and \(y\) are related by the differential equation \({{x}^{2}}\frac{\mathrm{d}y}{\mathrm{d}x}+xy=k\) for \(k\in \mathbb{R}\), show that \(y=\frac{k\left( \ln x+\alpha \right)}{x}\) is a solution of the differential equation where \(\alpha \) is an arbitrary constant.[2]

Hence, show that \(\left( {{\mathrm{e}}^{1-\alpha }},k{{\mathrm{e}}^{\alpha -1}} \right)\) is a stationary point of the curve \(y=\frac{k\left( \ln x+\alpha \right)}{x}\).[2]

By substituting \(v={{x}^{2}}+{{y}^{2}}\), show that the differential equation can be written as \(\frac{\mathrm{d}v}{\mathrm{d}x}=2\sqrt{v}\).[2]

Find \(v\) in terms of \(x\) and hence show that \({{y}^{2}}=\mathrm{f}\left( x \right)\) where \(\mathrm{f}\left( x \right)\) is to be determined.[4]