| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2010 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Substitution reducing to first order linear ODE |
| Difficulty | Standard +0.8 This is a Further Maths FP3 question requiring order reduction via substitution followed by integrating factor method. Part (a) is routine verification, but parts (b) and (c) require careful application of integrating factor technique and subsequent integration. While methodical rather than conceptually deep, it's above average difficulty due to being Further Maths content with multiple technical steps and potential for algebraic errors. |
| Spec | 4.10c Integrating factor: first order equations4.10d Second order homogeneous: auxiliary equation method |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\frac{du}{dx} + 2u = 3x \Rightarrow \frac{du}{dx} + \frac{u}{x} = 3\) | M1, A1 | CSO AG Substitution into LHS of DE and completion; 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| IF is \(\exp\left(\int \frac{k}{x}dx\right)\), for \(k = \pm 2, \pm 1\) and integration attempted | M1, A1;A1 | \(\exp\left(\int \frac{k}{x}dx\right)\) for \(k = \pm 2, \pm 1\) and integration attempted; 5 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{d}{dx}(ux^2) = 3x^2\) | M1 | LHS as differential of \(u \times\) IF |
| \(ux^2 = x^3 + A \Rightarrow u = x + Ax^{-2}\) | A1 | Must have an arbitrary constant |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = x + Ax^{-2} \Rightarrow y = \frac{1}{2}x^2 - \frac{A}{x} + B\) | M1, A1F | and with integration attempted; ft only if IF is M1A0A0; 2 marks total |
**(a)**
$u = \frac{dy}{dx} \Rightarrow \frac{du}{dx} = \frac{d^2y}{dx^2}$
$x\frac{du}{dx} + 2u = 3x \Rightarrow \frac{du}{dx} + \frac{u}{x} = 3$ | M1, A1 | CSO AG Substitution into LHS of DE and completion; 2 marks total
**(b)**
IF is $\exp\left(\int \frac{k}{x}dx\right)$, for $k = \pm 2, \pm 1$ and integration attempted | M1, A1;A1 | $\exp\left(\int \frac{k}{x}dx\right)$ for $k = \pm 2, \pm 1$ and integration attempted; 5 marks total
$= e^{2\ln x} \cdot = x^2$
$\frac{d}{dx}(ux^2) = 3x^2$ | M1 | LHS as differential of $u \times$ IF
$ux^2 = x^3 + A \Rightarrow u = x + Ax^{-2}$ | A1 | Must have an arbitrary constant
**(c)**
$\frac{dy}{dx} = x + Ax^{-2}$
$\frac{dy}{dx} = x + Ax^{-2} \Rightarrow y = \frac{1}{2}x^2 - \frac{A}{x} + B$ | M1, A1F | and with integration attempted; ft only if IF is M1A0A0; 2 marks total
**Total: 9 marks**
---3
\begin{enumerate}[label=(\alph*)]
\item A differential equation is given by
$$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 3 x$$
Show that the substitution
$$u = \frac { \mathrm { d } y } { \mathrm {~d} x }$$
transforms this differential equation into
$$\frac { \mathrm { d } u } { \mathrm {~d} x } + \frac { 2 } { x } u = 3$$
\item Find the general solution of
$$\frac { \mathrm { d } u } { \mathrm {~d} x } + \frac { 2 } { x } u = 3$$
giving your answer in the form $u = \mathrm { f } ( x )$.
\item Hence find the general solution of the differential equation
$$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 3 x$$
giving your answer in the form $y = \mathrm { g } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2010 Q3 [9]}}