| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Geometric curve properties |
| Difficulty | Standard +0.3 This is a straightforward differential equations question requiring separation of variables and using boundary conditions. The method is standard C4 material: set up dy/dx = k√y, separate variables, integrate (using standard result ∫y^(-1/2)dy = 2√y), apply initial conditions. While it requires multiple steps and careful algebraic manipulation, it follows a well-practiced procedure with no conceptual surprises or novel problem-solving required. Slightly easier than average due to the guided structure and standard technique. |
| Spec | 1.07b Gradient as rate of change: dy/dx notation1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks |
|---|---|
| \(\frac{dy}{dx} = k\sqrt{y}\) | M1 |
| \(\int y^{-\frac{1}{2}} \, dy = \int k \, dx\) | M1 |
| \(2y^{\frac{1}{2}} = kx + c\) | M1 A1 |
| \((0,4) \Rightarrow 4 = c\) | M1 |
| \(\therefore 2\sqrt{y} = kx + 4\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((2,9) \Rightarrow 6 = 2k+4, \quad k=1\) | M1 A1 | |
| \(2\sqrt{y} = x+4, \quad \sqrt{y} = \frac{1}{2}(x+4)\) | M1 | |
| \(y = \frac{1}{4}(x+4)^2\) | A1 | (9 marks) |
## (a)
$\frac{dy}{dx} = k\sqrt{y}$ | M1 |
$\int y^{-\frac{1}{2}} \, dy = \int k \, dx$ | M1 |
$2y^{\frac{1}{2}} = kx + c$ | M1 A1 |
$(0,4) \Rightarrow 4 = c$ | M1 |
$\therefore 2\sqrt{y} = kx + 4$ | A1 |
## (b)
$(2,9) \Rightarrow 6 = 2k+4, \quad k=1$ | M1 A1 |
$2\sqrt{y} = x+4, \quad \sqrt{y} = \frac{1}{2}(x+4)$ | M1 |
$y = \frac{1}{4}(x+4)^2$ | A1 | **(9 marks)**
---The gradient at any point $(x, y)$ on a curve is proportional to $\sqrt{y}$.
Given that the curve passes through the point with coordinates $(0, 4)$,
\begin{enumerate}[label=(\alph*)]
\item show that the equation of the curve can be written in the form
$$2\sqrt{y} = kx + 4,$$
where $k$ is a positive constant. [5]
\end{enumerate}
Given also that the curve passes through the point with coordinates $(2, 9)$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the equation of the curve in the form $y = \text{f}(x)$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q5 [9]}}