In this question, \(p\) denotes \(\dfrac{\d y}{\d x}\,\).
Given that
\[
y=p^2 +2 xp\,,
\]
show by differentiating with respect to \(x\) that
\[
\frac{\d x}{\d p} = -2 - \frac {2x} p .
\]
Hence show that \(x = -\frac23p +Ap^{-2}\,,\) where \(A\) is an arbitrary
constant.
Find \(y\) in terms of \(x\) if \(p=-3\) when \(x=2\).
Given instead that
\[ y=2xp +p \ln p\,,\]
and that \(p=1\) when \(x=-\frac14\), show that
\(x=-\frac12 \ln p - \frac14\,\) and find \(y\) in terms of \(x\).