Year: 2020
Paper: 3
Question Number: 7
Course: UFM Pure
Section: Second order differential equations
No solution available for this problem.
In spite of the change to criteria for entering the paper, there was still a very healthy number of candidates, and the vast majority handled the protocols for the online testing very well. Just over half the candidates attempted exactly six questions, and whilst about 10% attempted a seventh, hardly any did more than seven. With 20% attempting five questions, and 10% attempting only four, overall, there were very few candidates not attempting the target number. There was a spread of popularity across the questions, with no question attracting more than 90% of candidates and only one less than 10%, but every question received a good number of attempts. Likewise, there was a spread of success on the questions, though every question attracted at least one perfect solution.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item Given that the variables $x$, $y$ and $u$ are connected by the differential equations
\[ \frac{\mathrm{d}u}{\mathrm{d}x} + \mathrm{f}(x)u = \mathrm{h}(x) \quad \text{and} \quad \frac{\mathrm{d}y}{\mathrm{d}x} + \mathrm{g}(x)y = u, \]
show that
\[ \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (\mathrm{g}(x) + \mathrm{f}(x))\frac{\mathrm{d}y}{\mathrm{d}x} + (\mathrm{g}'(x) + \mathrm{f}(x)\mathrm{g}(x))y = \mathrm{h}(x). \tag{1} \]
\item Given that the differential equation
\[ \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + \left(1 + \frac{4}{x}\right)\frac{\mathrm{d}y}{\mathrm{d}x} + \left(\frac{2}{x} + \frac{2}{x^2}\right)y = 4x + 12 \tag{2} \]
can be written in the same form as (1), find a first order differential equation which is satisfied by $\mathrm{g}(x)$.
If $\mathrm{g}(x) = kx^n$, find a possible value of $n$ and the corresponding value of $k$.
Hence find a solution of (2) with $y = 5$ and $\dfrac{\mathrm{d}y}{\mathrm{d}x} = -3$ at $x = 1$.
\end{questionparts}
This was the second most popular question, but the most successful with a mean score of nearly two thirds marks. All but the weakest candidates managed to do part (I) perfectly well. Similarly, finding the first order differential equation for g(x) in part (ii) caused very few problems. Most candidates that attempted to substitute the given expression for g(x) in the first order differential equation obtained the correct polynomial equation, and a few gave up having done this. Most guessed the value n = -1 and then found that k = 2 works, whilst some just wrote the values of k and n, without any explanation. It wasn't uncommon for candidates to get stuck finding k or n, usually due to arithmetic errors. Most candidates attempting to find u(x) were able to find the integrating factor and perform the integration, although a significant proportion got the integral wrong. Regardless of accuracy, everyone attempted inserting the initial conditions. Some candidates also tried using a particular and complimentary solution method to integrate, but only a few who attempted that got the complimentary part correct. If candidates solved for u(x) correctly, they usually did so for y as well.