| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2024 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Standard +0.8 This is a Further Maths second-order differential equation requiring auxiliary equation solution, particular integral for exponential RHS, then applying two boundary conditions (one involving a limit at infinity, which constrains the complementary function). The final part requires solving a transcendental equation. More demanding than standard A-level but routine for Further Maths students who know the techniques. |
| Spec | 4.10a General/particular solutions: of differential equations4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| 14 | (a) | AE is 2 2 0 + − = |
| Answer | Marks |
|---|---|
| GS is y = A e − 2 x + B e x − 6 e − x | M1 |
| Answer | Marks |
|---|---|
| [7] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | Forming AE |
| Answer | Marks | Guidance |
|---|---|---|
| 14 | (b) | B=0 |
| Answer | Marks |
|---|---|
| x = − ln 2 | B1FT |
| Answer | Marks |
|---|---|
| [5] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | Equating coefficient(s) of their e𝑘𝑥 term(s) to 0 where |
Question 14:
14 | (a) | AE is 2 2 0 + − =
= −2, 1
CF is Ae −2x+Bex
PI is y = C e − x
y=−Ce −x , y=Ce −x (C−C−2C)e −x = 12e −x
C = − 6
GS is y = A e − 2 x + B e x − 6 e − x | M1
A1
A1
B1
M1
A1
A1
[7] | 1.1
1.1
1.1
1.1
2.1
1.1
1.1 | Forming AE
Attempt to differentiate their PI twice and substituting. Do
not condone 𝑦 = 𝐶e−2𝑥 or 𝑦 = 𝐶e𝑥.
Must see 𝑦 =
14 | (b) | B=0
d𝑦 = −2𝐴e−2𝑥 +𝐵e𝑥 +6e−𝑥
d𝑥
𝐴 = 3 (so y=3e −2x −6e −x )
y = 0 3 e − x ( e − x − 2 ) = 0 or 3−6e𝑥 = 0
x = − ln 2 | B1FT
M1*
A1
M1dep
A1
[5] | 3.1a
1.1
1.1
1.1
1.1 | Equating coefficient(s) of their e𝑘𝑥 term(s) to 0 where
𝑘 > 0
FT their GS. Accept d𝑦 = −2𝐴e−2𝑥 +6e−𝑥 if 𝐵 = 0
d𝑥
already found. Allow a slip. Not implied by
−2𝐴+𝐵+6 = 0.
Do not FT.
A complete method to solve their 𝑦 = 0 which leads to a
solution
or exact equivalent14
\begin{enumerate}[label=(\alph*)]
\item Find the general solution of the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } - 2 y = 12 e ^ { - x }$.
You are given that $y$ tends to zero as $x$ tends to infinity, and that $\frac { \mathrm { dy } } { \mathrm { dx } } = 0$ when $x = 0$.
\item Find the exact value of $x$ for which $y = 0$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2024 Q14 [12]}}