| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Standard +0.3 This is a standard second-order linear ODE with constant coefficients and a simple particular integral. Part (i) requires finding the complementary function (routine for auxiliary equation m²+4=0) and a particular integral using undetermined coefficients with sin x. Part (ii) applies initial conditions to find constants. While this is Further Maths content, it's a textbook exercise requiring only methodical application of standard techniques with no novel insight or tricky algebra. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((m^2 + 4 = 0) \Rightarrow m = \pm 2i\); CF = \(A\cos 2x + B\sin 2x\); PI = \(p\sin x(+ q\cos x)\); \(-p\sin(-q\cos x) + 4p\sin x(+4q\cos x) = \sin x\) \(\Rightarrow p = \frac{1}{3}, q = 0\); \(y = A\cos 2x + B\sin 2x + \frac{1}{3}\sin x\) | B1; B1; B1; M1; A1; B1√ 6 | For correct CF (AEtrig but not \(Ae^{2ix} + Be^{-2ix}\) only); State a trial PI with at least \(p\sin x\); For substituting PI into DE; For correct \(p\) and \(q\) (which may be implied); For using GS = CF + PI, with 2 arbitrary constants in CF and none in PI; For correct equation in A and/or B f.t. from their GS |
| (ii) \((0,0) \Rightarrow A = 0\); \(\frac{dy}{dx} = 2B\cos 2x + \frac{1}{3}\cos x = \frac{4}{3} = 2B + \frac{1}{3}\); \(A = 0, B = \frac{1}{2}\); \(y = \frac{1}{3}\sin 2x + \frac{1}{3}\sin x\) | B1√; M1; A1; A1 4 [10] | For correct equation in A and/or B f.t. from their GS; For differentiating their GS and substituting values for \(x\) and \(\frac{dy}{dx}\); For correct A and B. Allow \(A = -\frac{1}{4}i, B = \frac{1}{4}i\) from CF \(Ae^{2ix} + Be^{-2ix}\); For stating correct solution CAO |
**(i)** $(m^2 + 4 = 0) \Rightarrow m = \pm 2i$; CF = $A\cos 2x + B\sin 2x$; PI = $p\sin x(+ q\cos x)$; $-p\sin(-q\cos x) + 4p\sin x(+4q\cos x) = \sin x$ $\Rightarrow p = \frac{1}{3}, q = 0$; $y = A\cos 2x + B\sin 2x + \frac{1}{3}\sin x$ | B1; B1; B1; M1; A1; B1√ 6 | For correct CF (AEtrig but not $Ae^{2ix} + Be^{-2ix}$ only); State a trial PI with at least $p\sin x$; For substituting PI into DE; For correct $p$ and $q$ (which may be implied); For using GS = CF + PI, with 2 arbitrary constants in CF and none in PI; For correct equation in A and/or B f.t. from their GS
**(ii)** $(0,0) \Rightarrow A = 0$; $\frac{dy}{dx} = 2B\cos 2x + \frac{1}{3}\cos x = \frac{4}{3} = 2B + \frac{1}{3}$; $A = 0, B = \frac{1}{2}$; $y = \frac{1}{3}\sin 2x + \frac{1}{3}\sin x$ | B1√; M1; A1; A1 4 [10] | For correct equation in A and/or B f.t. from their GS; For differentiating their GS and substituting values for $x$ and $\frac{dy}{dx}$; For correct A and B. Allow $A = -\frac{1}{4}i, B = \frac{1}{4}i$ from CF $Ae^{2ix} + Be^{-2ix}$; For stating correct solution **CAO**
---\begin{enumerate}[label=(\roman*)]
\item Find the general solution of the differential equation
$$\frac{d^2y}{dx^2} + 4y = \sin x.$$ [6]
\item Find the solution of the differential equation for which $y = 0$ and $\frac{dy}{dx} = \frac{4}{3}$ when $x = 0$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2006 Q6 [10]}}