| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Standard linear first order - variable coefficients |
| Difficulty | Standard +0.8 This is a first-order linear differential equation requiring an integrating factor method (multiplying by x³), integration of 1/x⁴, and applying initial conditions. Part (ii) requires differentiation or analysis of the solution. While systematic, it's Further Maths content with multiple technical steps including handling negative powers, making it moderately harder than standard A-level calculus. |
| Spec | 4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (i) | dy 3 1 |
| Answer | Marks |
|---|---|
| 2x3 | M1 |
| Answer | Marks |
|---|---|
| [7] | 1.1 |
| Answer | Marks |
|---|---|
| 2.5 | Write equation in correct |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (ii) | e |
| Answer | Marks |
|---|---|
| and hence decreasing | B1 |
| Answer | Marks |
|---|---|
| [2] | 2.2a |
| 2.4 | Write y as sum of two |
Question 10:
10 | (i) | dy 3 1
(cid:14) y (cid:32)
dx x x2
integrating factor is exp(3 ln x)
integrating factor is exp(ln(x3)) = x3
dy
x3 (cid:14)3x2y(cid:32) x
dx
d (cid:11) x3y (cid:12)
(cid:32) x
dx
1
x3y (cid:32) x2 (cid:14)c
2
1
x(cid:32)1,y(cid:32)1(cid:159)c(cid:32)
2
1(cid:14)x2
y(cid:32)
2x3 | M1
A1
M1
A1A1
B1
c
A1
[7] | 1.1
1.1
2.1
m
1.1 1.1
1.1
i
2.5 | Write equation in correct
form (may be implied) and
attempt to find integrating
factor
Mnultiply by integrating
factor
e
A1 each side of equation
Use condition
Expressing y as a function of
x (can be in part (ii))
10 | (ii) | e
p
1 1
y = (cid:14)
2x3 2x
S
Explain that y is the sum of two decreasing functions
and hence decreasing | B1
E1
[2] | 2.2a
2.4 | Write y as sum of two
fractions OR differentiate
OR show that the derivative
is negative hence decreasing\begin{enumerate}[label=(\roman*)]
\item Obtain the solution to the differential equation
$$x \frac{dy}{dx} + 3y = \frac{1}{x}, \text{ where } x > 0,$$
given that $y = 1$ when $x = 1$. [7]
\item Deduce that $y$ decreases as $x$ increases. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core Q10 [9]}}