| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Asymptotic behavior for large values |
| Difficulty | Challenging +1.2 This is a standard FP2 second-order differential equation with repeated roots (auxiliary equation gives m=-3 twice) and a particular integral requiring the method of undetermined coefficients. Parts (a) and (b) are routine textbook exercises. Part (c) adds mild complexity by asking for asymptotic behavior, but students can recognize that for large t, the complementary function terms (involving e^{-3t}) become negligible, making the particular integral dominant—a straightforward application of standard technique with slightly elevated computational demand. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks |
|---|---|
| \(m^2 + 6m + 9 = 0\) | M1 |
| \(m = -3\) | A1 |
| C.F. \(x = (A + Bt)e^{-3t}\) | A1 |
| P.I. \(x = P\cos 3t + Q\sin 3t\) | B1 |
| \(\dot{x} = -3P\sin 3t + 3Q\cos 3t\) | M1 |
| \(\ddot{x} = -9P\cos 3t - 9Q\sin 3t\) | M1 |
| \((-9P\cos 3t - 9Q\sin 3t) + 6(-3P\sin 3t + 3Q\cos 3t) + 9(P\cos 3t + Q\sin 3t) = \cos t\) | M1 |
| \(-9P + 18Q + 9P = 1\) and \(-9Q - 18P + 9Q = 0\) | M1 |
| \(P = 0\) and \(Q = \frac{1}{18}\) | A1 |
| \(x = (A + Bt)e^{-3t} + \frac{1}{18}\sin 3t\) | A1ft |
| (8) |
| Answer | Marks | Guidance |
|---|---|---|
| \(t = 0: \quad x = A = \frac{1}{2}\) | B1 | |
| \(\dot{x} = -3(A + Bt)e^{-3t} + Be^{-3t} + \frac{3}{18}\cos 3t\) | M1 | |
| \(t = 0: \quad \dot{x} = -3A + B + \frac{1}{6} = 0\) | M1 A1 | \(B = \frac{4}{3}\) |
| \(x = \left(\frac{1}{2} + \frac{4t}{3}\right)e^{-3t} + \frac{1}{18}\sin 3t\) | A1 | |
| (5) |
| Answer | Marks | Guidance |
|---|---|---|
| \(t \approx \frac{59\pi}{6}\) | B1 | \((\approx 30.9)\) |
| \(x \approx -\frac{1}{18}\) | B1ft | |
| (2) 15 |
| Answer | Marks |
|---|---|
| 1st M1 Form auxiliary equation and correct attempt to solve. Can be implied from correct exponential. 2nd M1 for attempt to differentiate PI twice. 3rd M1 for substituting their expression into differential equation. 4th M1 for substitution of both boundary values. 1st M1 for correct attempt to differentiate their answer to part (a). 2nd M1 for substituting boundary value. |
**Part (a):**
| $m^2 + 6m + 9 = 0$ | M1 | |
| $m = -3$ | A1 | |
| C.F. $x = (A + Bt)e^{-3t}$ | A1 | |
| P.I. $x = P\cos 3t + Q\sin 3t$ | B1 | |
| $\dot{x} = -3P\sin 3t + 3Q\cos 3t$ | M1 | |
| $\ddot{x} = -9P\cos 3t - 9Q\sin 3t$ | M1 | |
| $(-9P\cos 3t - 9Q\sin 3t) + 6(-3P\sin 3t + 3Q\cos 3t) + 9(P\cos 3t + Q\sin 3t) = \cos t$ | M1 | |
| $-9P + 18Q + 9P = 1$ and $-9Q - 18P + 9Q = 0$ | M1 | |
| $P = 0$ and $Q = \frac{1}{18}$ | A1 | |
| $x = (A + Bt)e^{-3t} + \frac{1}{18}\sin 3t$ | A1ft | |
| | (8) | |
**Part (b):**
| $t = 0: \quad x = A = \frac{1}{2}$ | B1 | |
| $\dot{x} = -3(A + Bt)e^{-3t} + Be^{-3t} + \frac{3}{18}\cos 3t$ | M1 | |
| $t = 0: \quad \dot{x} = -3A + B + \frac{1}{6} = 0$ | M1 A1 | $B = \frac{4}{3}$ |
| $x = \left(\frac{1}{2} + \frac{4t}{3}\right)e^{-3t} + \frac{1}{18}\sin 3t$ | A1 | |
| | (5) | |
**Part (c):**
| $t \approx \frac{59\pi}{6}$ | B1 | $(\approx 30.9)$ |
| $x \approx -\frac{1}{18}$ | B1ft | |
| | (2) 15 | |
**Guidance Notes:**
| 1st M1 Form auxiliary equation and correct attempt to solve. Can be implied from correct exponential. 2nd M1 for attempt to differentiate PI twice. 3rd M1 for substituting their expression into differential equation. 4th M1 for substitution of both boundary values. 1st M1 for correct attempt to differentiate their answer to part (a). 2nd M1 for substituting boundary value. | |\begin{enumerate}
\item The differential equation
\end{enumerate}
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 x = \cos 3 t , \quad t \geqslant 0$$
describes the motion of a particle along the $x$-axis.\\
(a) Find the general solution of this differential equation.\\
(b) Find the particular solution of this differential equation for which, at $t = 0$,
$$x = \frac { 1 } { 2 } \text { and } \frac { \mathrm { d } x } { \mathrm {~d} t } = 0$$
On the graph of the particular solution defined in part (b), the first turning point for $t > 30$ is the point $A$.\\
(c) Find approximate values for the coordinates of $A$.\\
\hfill \mbox{\textit{Edexcel FP2 2011 Q8 [15]}}