| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Resonance cases requiring modified PI |
| Difficulty | Challenging +1.2 This is a standard resonance problem in FP2 requiring recognition that cos(8t) matches the complementary function frequency (ω=8), necessitating a modified particular integral of the form t(A sin 8t + B cos 8t). While it involves multiple parts including finding the general solution, applying initial conditions, and describing long-term behaviour, these are routine steps for Further Maths students who have learned the resonance case. The question is more procedural than conceptually challenging, though slightly above average difficulty due to the modified PI technique and multi-part nature. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(V = \lambda t\sin 8t\), \(\frac{dV}{dt} = \lambda\sin 8t + 8\lambda t\cos 8t\) | M1, A1 | |
| Substitute to give \(\frac{d^2V}{dt^2} = 16\lambda\cos 8t + 64\lambda t\sin 8t\) | A1 | |
| \(16\lambda\cos 8t = \cos 8t\), \(\therefore \lambda = \frac{1}{16}\) | M1, A1 (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| Auxiliary equation is \(m^2 + 64 = 0\), so \(m = \pm 8i\) | B1 | |
| Complementary function is \(A\cos 8t + B\sin 8t\) | M1 A1 | |
| General solution is \(A\cos 8t + B\sin 8t + \frac{1}{16}t\sin 8t\) | B1 (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(V=0\) when \(t=0\) implies \(A=0\) | ||
| \(8B\cos 8t + \frac{1}{16}\sin 8t + \frac{1}{2}t\cos 8t = 0\) when \(t=0\) | ||
| So \(8B = 0\) and \(V = \frac{1}{16}t\sin 8t\) is particular solution | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| As \(t\) becomes large the amplitude of the oscillations of \(V\) become large also. As \(t\to\infty\), \(V\to\infty\) also. | B1 (1) |
# Question 9:
## Part (a):
| Working | Marks | Notes |
|---------|-------|-------|
| $V = \lambda t\sin 8t$, $\frac{dV}{dt} = \lambda\sin 8t + 8\lambda t\cos 8t$ | M1, A1 | |
| Substitute to give $\frac{d^2V}{dt^2} = 16\lambda\cos 8t + 64\lambda t\sin 8t$ | A1 | |
| $16\lambda\cos 8t = \cos 8t$, $\therefore \lambda = \frac{1}{16}$ | M1, A1 (5) | |
## Part (b):
| Working | Marks | Notes |
|---------|-------|-------|
| Auxiliary equation is $m^2 + 64 = 0$, so $m = \pm 8i$ | B1 | |
| Complementary function is $A\cos 8t + B\sin 8t$ | M1 A1 | |
| General solution is $A\cos 8t + B\sin 8t + \frac{1}{16}t\sin 8t$ | B1 (4) | |
## Part (c):
| Working | Marks | Notes |
|---------|-------|-------|
| $V=0$ when $t=0$ implies $A=0$ | | |
| $8B\cos 8t + \frac{1}{16}\sin 8t + \frac{1}{2}t\cos 8t = 0$ when $t=0$ | | |
| So $8B = 0$ and $V = \frac{1}{16}t\sin 8t$ is particular solution | (3) | |
## Part (d):
| Working | Marks | Notes |
|---------|-------|-------|
| As $t$ becomes large the amplitude of the oscillations of $V$ become large also. As $t\to\infty$, $V\to\infty$ also. | B1 (1) | |9. Resonance in an electrical circuit is modelled by the differential equation
$$\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } } + 64 V = \cos 8 t$$
where $V$ represents the voltage in the circuit and $t$ represents time.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $\lambda$ for which $\lambda$ tsin8t is a particular integral of the differential equation.
\item Find the general solution of the differential equation.
Given that $V = 0$ and $\frac { \mathrm { d } V } { \mathrm {~d} t } = 0$ when $t = 0$,
\item find the particular solution of the equation.
\item Describe the behaviour of $V$ as $t$ becomes large, according to this model.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q9 [13]}}