Year: 2008
Paper: 3
Question Number: 6
Course: UFM Pure
Section: Second order differential equations
No solution available for this problem.
Most candidates attempted five, six or seven questions, and scored the majority of their total score on their best three or four. Those attempting seven or more tended not to do well, pursuing no single solution far enough to earn substantial marks.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
In this question, $p$ denotes $\dfrac{\d y}{\d x}\,$.
\begin{questionparts}
\item 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 $x=2$.
\item 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 $x$.
\end{questionparts}
More than 80% attempted this, and with more success than any other question. Having obtained the relation between x and p in each part, quite a few attempts then treated these as differential equations rather than merely substituting back to find expressions for y, and consequent inaccuracies lost marks.