Problem source

Show that, if $y^2 = x^k \f(x)$, 
then $\displaystyle 2xy \frac{\mathrm{d}y }{ \mathrm{d}x} = ky^2 + x^{k+1} 
\frac{\mathrm{d}\f }{ \mathrm{d}x}$\,. 
\begin{questionparts}
\item By setting $k=1$ in this result, find the solution of the differential equation
\[
\displaystyle 2xy \frac{\mathrm{d}y }{ \mathrm{d}x} = y^2 + x^2 - 1
\]
for which  $y=2$ when $x=1$. Describe geometrically this solution.
\item Find the solution of the differential equation 
\[
2x^2y\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x} = 2 \ln(x) - xy^2
\]
for which  $y=1$ when $x=1\,$.
\end{questionparts}