| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (polynomial/exponential x-side) |
| Difficulty | Moderate -0.3 This is a straightforward separable variables question with standard integration techniques. Part (a) requires separation, integrating x^(-2) and (1+2y)^(-1/2), then applying initial conditions. Part (b) is algebraic rearrangement to verify the given form. Slightly easier than average due to routine methods and verification rather than discovery. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int\frac{1}{\sqrt{1+2y}} dy = \int\frac{1}{x} dx\) | M1 | |
| \(\int\frac{1}{\sqrt{1+2y}} dy = k\sqrt{1+2y}\) | m1 | |
| \(\sqrt{1+2y} = -\frac{1}{x}(+c)\) | A1 | |
| \(x = 1, y = 4 \Rightarrow c = 4\) | m1 | |
| A1F | 6 marks | ft on \(k\) and \(\pm\frac{1}{x}\) only |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 + 2y = \left(4 - \frac{1}{x}\right)^2\) | m1 | |
| \(2y = 15 + \frac{1}{x^2} - \frac{8}{x}\) | A1 | 2 marks |
**8(a)**
| $\int\frac{1}{\sqrt{1+2y}} dy = \int\frac{1}{x} dx$ | M1 | | attempt to separate and integrate |
| $\int\frac{1}{\sqrt{1+2y}} dy = k\sqrt{1+2y}$ | m1 | | |
| $\sqrt{1+2y} = -\frac{1}{x}(+c)$ | A1 | | OE A1 for $-\frac{1}{x}$ depends on first M1 only |
| $x = 1, y = 4 \Rightarrow c = 4$ | m1 | | +c must be seen on previous line |
| | A1F | 6 marks | ft on $k$ and $\pm\frac{1}{x}$ only |
**8(b)**
| $1 + 2y = \left(4 - \frac{1}{x}\right)^2$ | m1 | | need $k\sqrt{1+2y} = 'x$ expression with + c' and attempt to square both sides |
| $2y = 15 + \frac{1}{x^2} - \frac{8}{x}$ | A1 | 2 marks | terms on RHS in any order; AG – be convinced CSO |
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**TOTAL MARKS: 75**8
\begin{enumerate}[label=(\alph*)]
\item Solve the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 + 2 y } } { x ^ { 2 } }$$
given that $y = 4$ when $x = 1$.
\item Show that the solution can be written as $y = \frac { 1 } { 2 } \left( 15 - \frac { 8 } { x } + \frac { 1 } { x ^ { 2 } } \right)$.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2007 Q8 [8]}}