| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2012 |
| 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 with h=0.1. Slightly above trivial due to being a two-part question requiring careful substitution, but easier than average A-level questions. |
| Spec | 1.09d Newton-Raphson method |
1 The function $y ( x )$ satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = \frac { y - x } { y ^ { 2 } + x }$$
and
$$y ( 1 ) = 2$$
\begin{enumerate}[label=(\alph*)]
\item Use the Euler formula
$$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$
with $h = 0.1$, to obtain an approximation to $y ( 1.1 )$.
\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 $y ( 1.2 )$, giving your answer to three decimal places.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2012 Q1 [6]}}