| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Standard linear first order - constant coefficients |
| Difficulty | Standard +0.8 This is a Further Maths FP3 question requiring the method of undetermined coefficients for a non-homogeneous first-order linear ODE. Part (a) demands substituting a trial solution into the DE and equating coefficients across multiple terms (polynomial and trigonometric), which is algebraically involved. Part (b) requires finding the complementary function and combining solutions. While systematic, this exceeds standard A-level in both technique sophistication and algebraic manipulation required. |
| Spec | 4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| PI: \(y_{PI} = a + bx + c\sin x + d\cos x\), \(y'_{PI} = b + c\cos x - d\sin x\) | ||
| \(b + c\cos x - d\sin x - 3a - 3bx - 3c\sin x - 3d\cos x = 10\sin x - 3x\) | M1 | Substituting into DE |
| \(b-3a=0;\ -3b=-3;\ c-3d=0;\ -d-3c=10\) | M1 | Equating coefficients (at least 2 equations) |
| \(a = \frac{1}{3};\ b=1;\ c=-3;\ d=-1\) | A2,1 | A1 for any two correct |
| \(y_{PI} = \frac{1}{3} + x - 3\sin x - \cos x\) | ||
| Total part | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Aux. eqn. \(m - 3 = 0\) | M1 | Altn. \(\int y^{-1}\,dy = \int 3\,dx\) OE (M1) |
| \(y_{CF} = Ae^{3x}\) | A1 | \(Ae^{3x}\) OE |
| \(y_{GS} = Ae^{3x} + \frac{1}{3} + x - 3\sin x - \cos x\) | B1F | 3 |
| Total | 7 |
## Question 2(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| PI: $y_{PI} = a + bx + c\sin x + d\cos x$, $y'_{PI} = b + c\cos x - d\sin x$ | | |
| $b + c\cos x - d\sin x - 3a - 3bx - 3c\sin x - 3d\cos x = 10\sin x - 3x$ | M1 | Substituting into DE |
| $b-3a=0;\ -3b=-3;\ c-3d=0;\ -d-3c=10$ | M1 | Equating coefficients (at least 2 equations) |
| $a = \frac{1}{3};\ b=1;\ c=-3;\ d=-1$ | A2,1 | A1 for any two correct |
| $y_{PI} = \frac{1}{3} + x - 3\sin x - \cos x$ | | |
| **Total part** | **4** | |
## Question 2(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Aux. eqn. $m - 3 = 0$ | M1 | Altn. $\int y^{-1}\,dy = \int 3\,dx$ OE (M1) |
| $y_{CF} = Ae^{3x}$ | A1 | $Ae^{3x}$ OE |
| $y_{GS} = Ae^{3x} + \frac{1}{3} + x - 3\sin x - \cos x$ | B1F | 3 | (c's CF + c's PI) with 1 arbitrary constant |
| **Total** | **7** | |
---2
\begin{enumerate}[label=(\alph*)]
\item Find the values of the constants $a , b , c$ and $d$ for which $a + b x + c \sin x + d \cos x$ is a particular integral of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } - 3 y = 10 \sin x - 3 x$$
(4 marks)
\item Hence find the general solution of this differential equation.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2008 Q2 [7]}}