| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Resonance cases requiring modified PI |
| Difficulty | Standard +0.3 This is a standard FP2 second-order differential equation question with repeated roots (auxiliary equation (r-3)²=0). Part (a) is routine verification of a particular integral, (b) requires standard complementary function plus particular integral, (c) applies initial conditions, and (d) involves basic curve sketching. All techniques are textbook exercises with no novel insight required, making it slightly easier than average even for Further Maths. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
$$\frac{d^2 y}{dt^2} - 6\frac{dy}{dt} + 9y = 4e^{3t}, \quad t \geq 0.$$
\begin{enumerate}[label=(\alph*)]
\item Show that $Kte^{3t}$ is a particular integral of the differential equation, where $K$ is a constant to be found. [4]
\item Find the general solution of the differential equation. [3]
Given that a particular solution satisfies $y = 3$ and $\frac{dy}{dt} = 1$ when $t = 0$,
\item find this solution. [4]
Another particular solution which satisfies $y = 1$ and $\frac{dy}{dt} = 0$ when $t = 0$, has equation
$$y = (1 - 3t + 2t^2)e^{3t}.$$
\item For this particular solution draw a sketch graph of $y$ against $t$, showing where the graph crosses the $t$-axis. Determine also the coordinates of the minimum of the point on the sketch graph. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q21 [16]}}