| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | Specimen |
| Marks | 6 |
| Topic | Differential equations |
| Type | Separable variables - standard (polynomial/exponential x-side) |
| Difficulty | Standard +0.3 This is a straightforward separable variables question requiring standard integration techniques (∫x^(-2)dx and ∫cos y dy), followed by applying an initial condition to find the constant. While it requires competent algebraic manipulation and making y the subject at the end, it follows a completely standard template with no conceptual challenges, making it slightly easier than average. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
Separate variables prior to integration **M1**
$\displaystyle\int \dfrac{1}{\sec y}\,\mathrm{d}y = \int \dfrac{1}{x^2}\,\mathrm{d}x$ **A1**
$\sin y = -\dfrac{1}{x}\ (+c)$ **A1** **A1**
Substitute in $y = \dfrac{\pi}{6}$ and $x = 4$ to get $c = \dfrac{3}{4}$ **M1**
$y = \sin^{-1}\!\left(\dfrac{3}{4} - \dfrac{1}{x}\right)$ o.e. **A1**
**Total: 6 marks**11 Solve the differential equation $x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \sec y$ given that $y = \frac { \pi } { 6 }$ when $x = 4$ giving your answer in the form $y = \mathrm { f } ( x )$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q11 [6]}}