| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | June |
| Marks | 18 |
| Topic | Second order differential equations |
| Type | Series solution from differential equation |
| Difficulty | Challenging +1.3 Part (a) requires substituting a given particular integral into a differential equation to find k, then combining with the complementary function—standard Further Maths technique. Part (b) involves implicit differentiation to find higher derivatives and constructing a Taylor series, which requires careful algebraic manipulation but follows a systematic procedure. While this is a substantial multi-part question requiring several Further Maths techniques, each step follows established methods without requiring novel insight. |
| Spec | 4.08a Maclaurin series: find series for function4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks |
|---|---|
| (b)(i) \(x=1,\ y=2\ \&\ \dfrac{\mathrm{d}y}{\mathrm{d}x}=1 \Rightarrow \dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\bigg | _{x=1} = -20\) B1 |
| Answer | Marks |
|---|---|
| \(\Rightarrow \dfrac{\mathrm{d}^3y}{\mathrm{d}x^3} + \left\{y^2\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} + 2y\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2\right\} + \left\{x\dfrac{\mathrm{d}y}{\mathrm{d}x} + y\right\} = 5 \Rightarrow \dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\bigg | _{x=1} = 78\) A1 A1 A1 |
**(a)** $y = kx\cos x \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = -kx\sin x + k\cos x$ and $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = -kx\cos x - 2k\sin x$
Attempt at 1st and 2nd derivatives using the Product Rule **M1**
Substituting both of these into the given DE **M1**
$-kx\cos x - 2k\sin x + kx\cos x = 4\sin x$
Comparing terms to evaluate $k$: $k = -2$ **M1 A1**
Aux. Eqn. $m^2 + 1 = 0$ solved $\Rightarrow m = \pm\mathrm{i}$ **M1 A1**
Comp. Fn. is $y_C = A\cos x + B\sin x$ **ft**. Accept $y_C = Ae^{\mathrm{i}x} + Be^{-\mathrm{i}x}$ here **B1**
G.S. is $y = A\cos x + B\sin x - 2x\cos x$ **ft** provided $y_P$ has no arb. consts. & $y_C$ has 2 **B1**
Do not accept final answer involving complex numbers
**[8]**
**(b)(i)** $x=1,\ y=2\ \&\ \dfrac{\mathrm{d}y}{\mathrm{d}x}=1 \Rightarrow \dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\bigg|_{x=1} = -20$ **B1**
Differentiating $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} + y^2\dfrac{\mathrm{d}y}{\mathrm{d}x} + xy = 5x - 19$ **M1**
Use of Product Rule and implicit differentiation (at least once) **M1**
$\Rightarrow \dfrac{\mathrm{d}^3y}{\mathrm{d}x^3} + \left\{y^2\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} + 2y\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2\right\} + \left\{x\dfrac{\mathrm{d}y}{\mathrm{d}x} + y\right\} = 5 \Rightarrow \dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\bigg|_{x=1} = 78$ **A1 A1 A1**
FT "78" from "−20" and also from $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ instead of $\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2$ (both = 1)
**[6]**
**(ii)** Use of $y = y(1) + (x-1)\cdot y'(1) + \frac{1}{2}(x-1)^2\cdot y''(1) + \frac{1}{6}(x-1)^3\cdot y'''(1) + \ldots$ **M1**
$= 2 + (x-1) - 10(x-1)^2 + 13(x-1)^3 + \ldots$ **ft** **A1**
Substituting $x = 1.1$ into this series $\Rightarrow y(1.1) \approx 2.013$ **ft** **M1 A1**
**[4]**10
\begin{enumerate}[label=(\alph*)]
\item Given that $y = k x \cos x$ is a particular integral for the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = 4 \sin x$$
determine the value of $k$ and find the general solution of this differential equation.
\item The variables $x$ and $y$ satisfy the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 5 x - 19$$
\begin{enumerate}[label=(\roman*)]
\item Given that $y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1$ when $x = 1$, find the value of $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$ when $x = 1$.
\item Deduce the Taylor series expansion for $y$ in ascending powers of $( x - 1 )$, up to and including the term in $( x - 1 ) ^ { 3 }$, and use this series to find an approximation correct to 3 decimal places for the value of $y$ when $x = 1.1$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q10 [18]}}