| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Standard +0.8 This is a Further Maths FP2 question requiring solution of a second-order linear ODE with complex roots in the complementary function, finding a particular integral for an exponential RHS, then applying two initial conditions. While systematic, it requires multiple techniques (auxiliary equation, particular integral method, applying ICs) and is inherently more demanding than standard A-level content. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(m^2 + 2m + 10 = 0 \Rightarrow m = \ldots\) | M1 | Form and solve the auxiliary equation |
| \(m = -1 \pm 3i\) | A1 | |
| \((y=)\ e^{-x}(A\cos 3x + B\sin 3x)\) or \(y = Ae^{(-1+3i)x} + Be^{(-1-3i)x}\) | A1 | \(y=\) not needed. May be seen with \(\theta\) instead of \(x\) |
| \(y = ke^{-x},\ y' = -ke^{-x},\ y'' = ke^{-x}\) | M1 | \(y = ke^{-x}\) and attempt to differentiate twice |
| \(e^{-x}(k - 2k + 10k) = 27e^{-x} \Rightarrow k = 3\) | A1 | |
| \(y = e^{-x}(A\cos 3x + B\sin 3x + 3)\) or \(y = Ae^{(-1+3i)x} + Be^{(-1-3i)x} + 3e^{-x}\) | B1ft (6) | Must have \(x\) and \(y=\). NB A1 on e-ft, so \(y =\) their CF + their PI |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x=0,\ y=0 \Rightarrow A = (-3)\) | M1 | Uses \(x=0\), \(y=0\) in attempt to find \(A\) |
| \(y' = -e^{-x}(A\cos 3x + B\sin 3x + 3) + e^{-x}(3B\cos 3x - 3A\sin 3x)\) | M1A1 | M1: Attempt to differentiate using product rule with \(A\) or their value of \(A\). A1: Correct derivative with \(A\) or their value |
| \(x=0,\ y'=0 \Rightarrow B = 0\) | M1A1 | M1: Uses \(x=0\), \(y'=0\) and their value of \(A\) to find \(B\). A1: \(B=0\) |
| \(y = e^{-x}(3 - 3\cos 3x)\) | A1 (6) | cao and cso |
# Question 5:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $m^2 + 2m + 10 = 0 \Rightarrow m = \ldots$ | M1 | Form and solve the auxiliary equation |
| $m = -1 \pm 3i$ | A1 | |
| $(y=)\ e^{-x}(A\cos 3x + B\sin 3x)$ or $y = Ae^{(-1+3i)x} + Be^{(-1-3i)x}$ | A1 | $y=$ not needed. May be seen with $\theta$ instead of $x$ |
| $y = ke^{-x},\ y' = -ke^{-x},\ y'' = ke^{-x}$ | M1 | $y = ke^{-x}$ and attempt to differentiate twice |
| $e^{-x}(k - 2k + 10k) = 27e^{-x} \Rightarrow k = 3$ | A1 | |
| $y = e^{-x}(A\cos 3x + B\sin 3x + 3)$ or $y = Ae^{(-1+3i)x} + Be^{(-1-3i)x} + 3e^{-x}$ | B1ft (6) | Must have $x$ and $y=$. NB A1 on e-ft, so $y =$ their CF + their PI |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $x=0,\ y=0 \Rightarrow A = (-3)$ | M1 | Uses $x=0$, $y=0$ in attempt to find $A$ |
| $y' = -e^{-x}(A\cos 3x + B\sin 3x + 3) + e^{-x}(3B\cos 3x - 3A\sin 3x)$ | M1A1 | M1: Attempt to differentiate using product rule with $A$ or their value of $A$. A1: Correct derivative with $A$ or their value |
| $x=0,\ y'=0 \Rightarrow B = 0$ | M1A1 | M1: Uses $x=0$, $y'=0$ and their value of $A$ to find $B$. A1: $B=0$ |
| $y = e^{-x}(3 - 3\cos 3x)$ | A1 (6) | cao and cso |\begin{enumerate}
\item (a) Find the general solution of the differential equation
\end{enumerate}
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 27 \mathrm { e } ^ { - x }$$
(b) Find the particular solution that satisfies $y = 0$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 0$\\
\hfill \mbox{\textit{Edexcel FP2 2014 Q5 [12]}}