Year: 2009
Paper: 3
Question Number: 7
Course: UFM Pure
Section: Second order differential equations
No solution available for this problem.
The vast majority of candidates (in excess of 95%) attempted at least five questions, and nearly a quarter attempted more than six questions, though very few doing so achieved high scores (about 2%). Most attempting more than six questions were submitting fragmentary answers, which, as the rubric informed candidates, earned little credit.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1485.5
Banger Comparisons: 1
\begin{questionparts}
\item
The functions
$\f_n(x)$ are defined for $n=0$, $1$, $2$, $\ldots$\, , by
\[
\f_0(x) = \frac 1 {1+x^2}\,
\qquad \text{and}\qquad
\f_{n+1}(x) =\frac{\d \f_n(x)}{\d x}\,.
\]
Prove, for $n\ge1$, that
\[
(1+x^2)\f_{n+1}(x) + 2(n+1)x\f_n(x) + n(n+1)\f_{n-1}(x)=0\,.
\]
\item
The functions $\P_n(x)$ are defined for $n=0$, $1$, $2$, $\ldots$\, , by
\[
\P_n(x) = (1+x^2)^{n+1}\f_n(x)\,.
\]
Find expressions for $\P_0(x)$, $\P_1(x)$ and $\P_2(x)$.
Prove, for $n\ge0$, that
\[
\P_{n+1}(x) -(1+x^2)\frac {\d \P_n(x)}{\d x}+ 2(n+1)x \P_n(x)=0\,,
\]
and that $\P_n(x)$ is a polynomial of degree $n$.
\end{questionparts}
Approximately two thirds of the candidates attempted this, earning roughly half marks in doing so. Part (i) and finding the three expressions for P₀, P₁ & P₂ from part (ii) largely went well. The result involving Pₙ₊₁ saw most falling by the wayside, especially those who attempted it by induction. Quite a few candidates did score all but two marks in proving that Pₙ was a polynomial of degree n or less, but not appreciating that there was still something to do regarding the leading term.