| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Apply basic Euler method for differential equations |
| Difficulty | Standard +0.3 This is a straightforward application of two standard numerical methods (Euler's method and midpoint method) with given formulas. Students simply substitute values into provided formulas with no derivation, problem-solving, or conceptual insight required. The arithmetic is routine, making this slightly easier than average for Further Maths FP3. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
1 It is given that $y ( x )$ satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = \sqrt { 2 x + y }$$
and
$$y ( 3 ) = 5$$
\begin{enumerate}[label=(\alph*)]
\item Use the Euler formula
$$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$
with $h = 0.2$, to obtain an approximation to $y ( 3.2 )$, giving your answer to four decimal places.
\item Use the formula
$$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$
with your answer to part (a), to obtain an approximation to $y ( 3.4 )$, giving your answer to three decimal places.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2013 Q1 [6]}}