| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2007 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Substitution reducing to first order linear ODE |
| Difficulty | Hard +2.3 This AEA question requires sophisticated insight to recognize that the substitution h(x) = (dy/dx)² transforms a functional equation into a differential equation, followed by solving a first-order DE with integrating factor technique. The novel setup, multiple conceptual leaps, and extended multi-step reasoning place it well above typical A-level questions. |
| Spec | 1.07b Gradient as rate of change: dy/dx notation1.08k Separable differential equations: dy/dx = f(x)g(y) |
4.The function $\mathrm { h } ( x )$ has domain $\mathbb { R }$ and range $\mathrm { h } ( x ) > 0$ ,and satisfies
$$\sqrt { \int \mathrm { h } ( x ) \mathrm { d } x } = \int \sqrt { \mathrm { h } ( x ) } \mathrm { d } x$$
\begin{enumerate}[label=(\alph*)]
\item By substituting $\mathrm { h } ( x ) = \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 }$ ,show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( y + c ) ,$$
where $c$ is constant.
\item Hence find a general expression for $y$ in terms of $x$ .
\item Given that $\mathrm { h } ( 0 ) = 1$ ,find $\mathrm { h } ( x )$ .
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2007 Q4 [11]}}