| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Session | Specimen |
| Marks | 14 |
| Topic | Differential equations |
| Type | Separable variables - standard (polynomial/exponential x-side) |
| Difficulty | Standard +0.8 This question requires separable differential equations with integration involving trigonometric substitution (part a), implicit differentiation with constraint analysis requiring completing the square (part b(i)), and second-order implicit differentiation (part b(ii)). While the techniques are A-level standard, the combination of non-trivial integration, algebraic manipulation to establish an inequality condition, and careful implicit differentiation across multiple parts makes this moderately harder than average. |
| Spec | 1.07s Parametric and implicit differentiation1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) (a) Rearrange in separated form \(\frac{1}{y}\frac{dy}{dx} = \frac{1 + x\cos x}{x}\) | B1 | Apply integration with respect to \(x\) to both sides: \(\int\frac{1}{y}dy = \int\left(\frac{1}{x} + \cos x\right)dx\) (or \(\int\left(\frac{1}{x} + \frac{\cos x}{\sin x}\right)dx\)) |
| (b) Apply initial condition to obtain \(A\) | M1 | Obtain \(y = \frac{2}{\pi}x\sin x\) |
| (ii) (a) Differentiate implicitly with respect to \(x\) | M1 | \(2x - 2y\frac{dy}{dx} = 2a\) |
| (b) Differentiate implicitly | M1 | \(2 - 2\frac{dy}{dx} \cdot \frac{dy}{dx} - 2y\frac{d^2y}{dx^2} = 0\) and rearrange to obtain displayed result |
(i) (a) Rearrange in separated form $\frac{1}{y}\frac{dy}{dx} = \frac{1 + x\cos x}{x}$ | B1 | Apply integration with respect to $x$ to both sides: $\int\frac{1}{y}dy = \int\left(\frac{1}{x} + \cos x\right)dx$ (or $\int\left(\frac{1}{x} + \frac{\cos x}{\sin x}\right)dx$) | M1 | Obtain $\ln y = \ln x + \ln \sin x + c$ | A1A1 | Obtain $y = Ax \sin x$ | A1 | 5 marks
(b) Apply initial condition to obtain $A$ | M1 | Obtain $y = \frac{2}{\pi}x\sin x$ | A1 | 2 marks
(ii) (a) Differentiate implicitly with respect to $x$ | M1 | $2x - 2y\frac{dy}{dx} = 2a$ | A1 | $\frac{dy}{dx} = 0 \Rightarrow x = a$ | M1 | Substitute $x = a \Rightarrow y^2 = b - a^2$ | M1 | $y$ is real if $b \geq a^2$ | A1 (AG) | 5 marks
(b) Differentiate implicitly | M1 | $2 - 2\frac{dy}{dx} \cdot \frac{dy}{dx} - 2y\frac{d^2y}{dx^2} = 0$ and rearrange to obtain displayed result | A1 (AG) | 2 marks\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the general solution of the differential equation
$$x \frac{dy}{dx} = y(1 + x \cot x),$$
expressing $y$ in terms of $x$. [5]
\item Find the particular solution given that $y = 1$ when $x = \frac{1}{2}\pi$. [2]
\end{enumerate}
\item The real variables $x$ and $y$ are related by $x^2 - y^2 = 2ax - b$, where $a$ and $b$ are real constants.
\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{dy}{dx} = 0$ can only be solved for $x$ and $y$ if $b \geqslant a^2$. [5]
\item Show that $y \frac{d^2y}{dx^2} = 1 - \left(\frac{dy}{dx}\right)^2$. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 Q8 [14]}}