| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 7 |
| 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 technique: separate variables, integrate both sides (using power rule for √x and simple reciprocal for 1/y²), apply initial condition, and rearrange to make y the subject. While it requires multiple steps, the integration is routine and the method is a core C4 skill with no conceptual challenges. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int \frac{1}{y^2} \, dy = \int \sqrt{x} \, dx\) | M1 | |
| \(-y^{-1} = \frac{2}{3} x^{\frac{3}{2}} + c\) | M1 A1 | |
| \(x = 1, y = -2 \Rightarrow -\frac{1}{2} = \frac{2}{3} + c, \quad c = -\frac{1}{6}\) | M1 A1 | |
| \(-\frac{1}{y} = \frac{2}{3} x^{\frac{3}{2}} - \frac{1}{6}, \quad \frac{1}{y} = \frac{1}{6} - \frac{2}{3} x^{\frac{3}{2}} = \frac{1}{6}(1 - 4x^{\frac{3}{2}})\) | M1 | |
| \(y = \frac{6}{1 - 4x^{\frac{3}{2}}}\) | A1 | (7) |
$\int \frac{1}{y^2} \, dy = \int \sqrt{x} \, dx$ | M1 |
$-y^{-1} = \frac{2}{3} x^{\frac{3}{2}} + c$ | M1 A1 |
$x = 1, y = -2 \Rightarrow -\frac{1}{2} = \frac{2}{3} + c, \quad c = -\frac{1}{6}$ | M1 A1 |
$-\frac{1}{y} = \frac{2}{3} x^{\frac{3}{2}} - \frac{1}{6}, \quad \frac{1}{y} = \frac{1}{6} - \frac{2}{3} x^{\frac{3}{2}} = \frac{1}{6}(1 - 4x^{\frac{3}{2}})$ | M1 |
$y = \frac{6}{1 - 4x^{\frac{3}{2}}}$ | A1 | **(7)**\begin{enumerate}
\item Given that $y = - 2$ when $x = 1$, solve the differential equation
\end{enumerate}
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ^ { 2 } \sqrt { x }$$
giving your answer in the form $y = \mathrm { f } ( x )$.\\
\hfill \mbox{\textit{Edexcel C4 Q2 [7]}}