| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Asymptotic behavior for large values |
| Difficulty | Standard +0.8 This is a standard second-order linear ODE with constant coefficients requiring the complementary function (complex roots: -1±4i) and particular integral (linear form). While methodical, it's Further Maths content with complex numbers in the auxiliary equation, multiple algebraic steps, and requires understanding asymptotic behavior—moderately above average difficulty. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Aux. equation \(m^2 + 2m + 17 = 0\) | M1 | For attempting to solve correct auxiliary equation |
| \(\Rightarrow m = -1 \pm 4i\) | A1 | For correct roots |
| CF \((y = e^{-x}(A\cos 4x + B\sin 4x)\) | A1 V | For correct CF (allow \(A\frac{\cos}{sin}(4x + c)\)) (trig terms required, not \(e^{±4i x}\)) |
| f.t. from their \(m\) with 2 arbitrary constants | ||
| PI \((y = px + q \Rightarrow 2p + 17(px + q) = 17x + 36\) | M1 | For stating and substituting PI of correct form |
| \(\Rightarrow p = 1\) and \(q = 2\) | A1 | For correct value of \(p\) |
| A1 | For correct value of \(q\) | |
| GS \(y = e^{-x}(A\cos 4x + B\sin 4x) + x + 2\) | B1 7 | For GS, f.t. from their CF+PI with 2 arbitrary constants in CF and none in PI. Requires \(\boxed{y = }\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x \to 0 \Rightarrow e^{-x} \to 0\) OR very small | B1 | For correct statement. Allow graph |
| \(\Rightarrow y = x + 2\) approximately | B1 V 2 | For correct equation |
| Allow \(=\), \(\to\) and in words. Allow relevant f.t. from linear part of GS |
## (i)
Aux. equation $m^2 + 2m + 17 = 0$ | M1 | For attempting to solve correct auxiliary equation
$\Rightarrow m = -1 \pm 4i$ | A1 | For correct roots
CF $(y = e^{-x}(A\cos 4x + B\sin 4x)$ | A1 V | For correct CF (allow $A\frac{\cos}{sin}(4x + c)$) (trig terms required, not $e^{±4i x}$)
| | f.t. from their $m$ with 2 arbitrary constants
PI $(y = px + q \Rightarrow 2p + 17(px + q) = 17x + 36$ | M1 | For stating and substituting PI of correct form
$\Rightarrow p = 1$ and $q = 2$ | A1 | For correct value of $p$
| A1 | For correct value of $q$
GS $y = e^{-x}(A\cos 4x + B\sin 4x) + x + 2$ | B1 7 | For GS, f.t. from their CF+PI with 2 arbitrary constants in CF and none in PI. Requires $\boxed{y = }$
## (ii)
$x \to 0 \Rightarrow e^{-x} \to 0$ OR very small | B1 | For correct statement. Allow graph
$\Rightarrow y = x + 2$ approximately | B1 V 2 | For correct equation
| | Allow $=$, $\to$ and in words. Allow relevant f.t. from linear part of GS
---\begin{enumerate}[label=(\roman*)]
\item Find the general solution of the differential equation
$$\frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + 17y = 17x + 36.$$ [7]
\item Show that, when $x$ is large and positive, the solution approximates to a linear function, and state its equation. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2010 Q6 [9]}}