| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| 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 to find a particular integral, then solving using integrating factor and applying initial conditions. While the techniques are standard for Further Maths, the combination of finding a particular integral with trigonometric terms, computing the complementary function, and applying boundary conditions makes this moderately challenging, requiring careful algebraic manipulation across multiple steps. |
| Spec | 4.10c Integrating factor: first order equations |
## Question 2(a)
[4 marks]
M1: Substitute $y = a + b\sin 2x + c\cos 2x$ into differential equation
M1: Find $\frac{dy}{dx} = 2b\cos 2x - 2c\sin 2x$
M1: Equate coefficients of constant, $\sin 2x$ and $\cos 2x$ terms
A1: Correct values $a = 5$, $b = 0$, $c = -5$
## Question 2(b)
[4 marks]
M1: Find complementary function $y_c = Ae^{-4x}$
M1: State particular integral $y_p = 5 - 5\cos 2x$
M1: General solution $y = Ae^{-4x} + 5 - 5\cos 2x$
A1: Apply initial condition $y = 4$ when $x = 0$ to find $A = -1$
A1: Final solution $y = -e^{-4x} + 5 - 5\cos 2x$2
\begin{enumerate}[label=(\alph*)]
\item Find the values of the constants $a$, $b$ and $c$ for which $a + b \sin 2 x + c \cos 2 x$ is a particular integral of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 4 y = 20 - 20 \cos 2 x$$
[4 marks]
\item Hence find the solution of this differential equation, given that $y = 4$ when $x = 0$.\\[0pt]
[4 marks]
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-04_1974_1709_733_153}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2014 Q2 [8]}}