| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 10 |
| Topic | First order differential equations (integrating factor) |
| Type | Standard linear first order - variable coefficients |
| Difficulty | Challenging +1.3 This is a first-order linear ODE requiring an integrating factor (μ = x²) followed by partial fraction decomposition of a rational function with four factors. While the technique is standard Further Maths fare, the algebraic complexity of the partial fractions and subsequent integration of multiple logarithmic and arctangent terms makes this more demanding than a typical C4/FP1 question, though not exceptionally difficult for Further Maths students who have practiced these methods. |
| Spec | 4.05c Partial fractions: extended to quadratic denominators4.10c Integrating factor: first order equations |
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of the differential equation
$$\frac{dy}{dx} + \frac{2y}{x} = \frac{x+3}{x(x-1)(x^2+3)} \quad (x > 1)$$
[8]
\item Find the particular solution for which $y = 0$ when $x = 3$.
Give your answer in the form $y = f(x)$.
[2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q10 [10]}}