| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (polynomial/exponential x-side) |
| Difficulty | Standard +0.8 This is a separable differential equation requiring manipulation of trigonometric functions, integration of sec²(2x) (needing substitution or recognition), and finding a particular solution. While the separation is straightforward, the integration steps and algebraic manipulation of the constant exceed typical C4 routine exercises, placing it moderately above average difficulty. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Separate variables as \(\sec^2 y \, dy = 2 | \cos^2 2x \, dx\) | M1 |
| LHS \(= \tan y\) | A1 | |
| RHS; attempt to change to double angle | M1 | |
| Correctly shown as \(1 + \cos 4x\) | A1 | |
| \(\int \cos 4x \, dx = \frac{1}{4} \sin 4x\) | A1 | |
| Completely correct equation (other than +c) | A1 | 7 marks |
| +c on either side | A1 | |
| (ii) Use boundary condition | M1 | |
| c (on RHS) \(= 1\) | A1 | |
| Substitute \(x = \frac{1}{8}\pi\) into their eqn, produce \(y = 1.05\) | A1 | 3 marks |
(i) Separate variables as $\sec^2 y \, dy = 2| \cos^2 2x \, dx$ | M1 | seen or implied
LHS $= \tan y$ | A1 |
RHS; attempt to change to double angle | M1 |
Correctly shown as $1 + \cos 4x$ | A1 |
$\int \cos 4x \, dx = \frac{1}{4} \sin 4x$ | A1 |
Completely correct equation (other than +c) | A1 | 7 marks | $\tan y = x + \frac{1}{4} \sin 4x$; not on both sides unless $c_1$ and $c_2$ provided, a sensible outcome would ensue or $c_2 - c_1 = 1$; not fortuitously obtained
+c on either side | A1 |
(ii) Use boundary condition | M1 |
c (on RHS) $= 1$ | A1 |
Substitute $x = \frac{1}{8}\pi$ into their eqn, produce $y = 1.05$ | A1 | 3 marks | or 4.19 or 7.33 etc. Radians only
---\begin{enumerate}[label=(\roman*)]
\item Find the general solution of the differential equation
$$\frac{\sec^2 y}{\cos^2(2x)} \frac{dy}{dx} = 2.$$ [7]
\item For the particular solution in which $y = \frac{1}{4}\pi$ when $x = 0$, find the value of $y$ when $x = \frac{1}{8}\pi$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C4 2007 Q9 [10]}}