| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 7 |
| Topic | First order differential equations (integrating factor) |
| Type | Applied/modelling contexts |
| Difficulty | Standard +0.8 This is a first-order linear ODE with non-trivial right-hand side requiring integrating factor method and integration by parts (likely twice for the cos t term with exponential). While the method is standard for Further Maths, the algebraic manipulation and integration steps are more involved than typical A-level questions, placing it moderately above average difficulty. |
| Spec | 4.10c Integrating factor: first order equations |
A particular radioactive substance decays over time.
A scientist models the amount of substance, $x$ grams, at time $t$ hours by the differential equation
$$\frac{dx}{dt} + \frac{1}{10}x = e^{-0.1t}\cos t.$$
\begin{enumerate}[label=(\roman*)]
\item Solve the differential equation to find the general solution for $x$ in terms of $t$. [3]
\end{enumerate}
Initially there was $10$ g of the substance.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the particular solution of the differential equation. [2]
\item Find to $6$ significant figures the amount of substance that would be predicted by the model at
\begin{enumerate}[label=(\alph*)]
\item $6$ hours, [1]
\item $6.25$ hours. [1]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q10 [7]}}