| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| 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 requiring standard integration techniques (∫y dy and ∫sec²x dx) with a simple initial condition substitution. While it requires competent execution of the separation method and knowledge of standard integrals, it's a routine textbook exercise with no conceptual challenges, making it slightly easier than the average A-level question. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int y\, dy = \int \frac{3}{\cos^2 x}\, dx = \int 3\sec^2 x\, dx\) | B1 | Can be implied. Ignore integral signs |
| \(\frac{1}{2}y^2 = 3\tan x \; (+C)\) | M1 A1 | |
| Using \(y = 2,\, x = \frac{\pi}{4}\): \(\frac{1}{2}(2)^2 = 3\tan\frac{\pi}{4} + C\) | M1 | |
| \(C = -1\) | ||
| \(\frac{1}{2}y^2 = 3\tan x - 1\) | A1 (5) | or equivalent |
## Question 4:
| $\int y\, dy = \int \frac{3}{\cos^2 x}\, dx = \int 3\sec^2 x\, dx$ | B1 | Can be implied. Ignore integral signs |
| $\frac{1}{2}y^2 = 3\tan x \; (+C)$ | M1 A1 | |
| Using $y = 2,\, x = \frac{\pi}{4}$: $\frac{1}{2}(2)^2 = 3\tan\frac{\pi}{4} + C$ | M1 | |
| $C = -1$ | | |
| $\frac{1}{2}y^2 = 3\tan x - 1$ | A1 (5) | or equivalent |
---\begin{enumerate}
\item Given that $y = 2$ at $x = \frac { \pi } { 4 }$, solve the differential equation
\end{enumerate}
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { y \cos ^ { 2 } x }$$
\hfill \mbox{\textit{Edexcel C4 2012 Q4 [5]}}