| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 10 |
| 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 structured Further Maths question requiring multiple techniques: order reduction by substitution, integrating factor method, and integration back to find y. While part (a) is routine verification and part (b) is standard integrating factor application, the multi-step nature and Further Maths context place it moderately above average difficulty. The question guides students through each stage, preventing it from being truly challenging. |
| Spec | 4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(u = \dfrac{dy}{dx} \Rightarrow \dfrac{du}{dx} = \dfrac{d^2y}{dx^2}\) | M1 | |
| \(x\dfrac{du}{dx} - u = 3x^2 \Rightarrow \dfrac{du}{dx} - \dfrac{1}{x}u = 3x\) | A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| IF is \(\exp\!\left(\int -\dfrac{1}{x}\,dx\right)\) | M1 | and with integration attempted |
| \(= e^{-\ln x}\) | A1 | |
| \(= x^{-1}\) or \(\dfrac{1}{x}\) | A1 | or multiple of \(x^{-1}\) |
| \(\dfrac{d}{dx}\!\left[ux^{-1}\right] = 3\) | M1 | LHS as differential of \(u \times\) IF. PI |
| \(\Rightarrow ux^{-1} = 3x + A\) | m1 | Must have arbitrary constant (dep. on previous M1 only) |
| \(u = 3x^2 + Ax\) | A1 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\dfrac{dy}{dx} = 3x^2 + Ax\) | M1 | Replaces \(u\) by \(\dfrac{dy}{dx}\) and attempts to integrate |
| \(y = x^3 + \dfrac{Ax^2}{2} + B\) | A1F | 2 |
| Total | 10 |
## Question 4(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $u = \dfrac{dy}{dx} \Rightarrow \dfrac{du}{dx} = \dfrac{d^2y}{dx^2}$ | M1 | |
| $x\dfrac{du}{dx} - u = 3x^2 \Rightarrow \dfrac{du}{dx} - \dfrac{1}{x}u = 3x$ | A1 | 2 | AG — substitution into LHS of DE and completion |
## Question 4(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| IF is $\exp\!\left(\int -\dfrac{1}{x}\,dx\right)$ | M1 | and with integration attempted |
| $= e^{-\ln x}$ | A1 | |
| $= x^{-1}$ or $\dfrac{1}{x}$ | A1 | or multiple of $x^{-1}$ |
| $\dfrac{d}{dx}\!\left[ux^{-1}\right] = 3$ | M1 | LHS as differential of $u \times$ IF. PI |
| $\Rightarrow ux^{-1} = 3x + A$ | m1 | Must have arbitrary constant (dep. on previous M1 only) |
| $u = 3x^2 + Ax$ | A1 | 6 | |
## Question 4(c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\dfrac{dy}{dx} = 3x^2 + Ax$ | M1 | Replaces $u$ by $\dfrac{dy}{dx}$ and attempts to integrate |
| $y = x^3 + \dfrac{Ax^2}{2} + B$ | A1F | 2 | ft on cand's $u$ but solution must have two arbitrary constants |
| **Total** | **10** | |
---4
\begin{enumerate}[label=(\alph*)]
\item A differential equation is given by
$$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 }$$
Show that the substitution
$$u = \frac { \mathrm { d } y } { \mathrm {~d} x }$$
transforms this differential equation into
$$\frac { \mathrm { d } u } { \mathrm {~d} x } - \frac { 1 } { x } u = 3 x$$
\item By using an integrating factor, find the general solution of
$$\frac { \mathrm { d } u } { \mathrm {~d} x } - \frac { 1 } { x } u = 3 x$$
giving your answer in the form $u = \mathrm { f } ( x )$.
\item Hence find the general solution of the differential equation
$$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 }$$
giving your answer in the form $y = \mathrm { g } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2008 Q4 [10]}}