| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Standard non-homogeneous with polynomial RHS |
| Difficulty | Standard +0.8 This is a standard second-order linear non-homogeneous differential equation requiring both complementary function (solving auxiliary equation with complex roots) and particular integral (trying polynomial form). While the method is systematic, it involves multiple steps including complex number manipulation, coefficient matching for a quadratic particular integral, and interpreting asymptotic behavior. This is typical Further Maths content but more involved than average A-level questions. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| \(6m^2 + 5m + 1 = 0 \Rightarrow m = -\frac{1}{2}, -\frac{1}{3}\) | M1 | Auxiliary equation |
| \(x = Ae^{-\frac{1}{2}t} + Be^{-\frac{1}{3}t}\) | A1 | Complementary function. Allow with '\(x=\)' missing |
| \(x = p + qt + rt^2 \Rightarrow x' = q + 2rt \Rightarrow x'' = 2r\) | B1 | Particular integral and its derivatives |
| \(12r + 5q + 10rt + p + qt + rt^2 = t^2 + 10t + 13\) | M1 | Substitutes and equates coefficients |
| \(r = 1,\quad q = 0,\quad p = 1\) | A1 | |
| \(x = Ae^{-\frac{1}{2}t} + Be^{-\frac{1}{3}t} + t^2 + 1\) | A1 | Must have \(x=\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = t^2 + 1\) | B1FT | Must have '\(x=\)'. Accept '\(x\) is approximately' but do not accept '\(x \rightarrow\)'. FT on their PI |
## Question 2(a):
| $6m^2 + 5m + 1 = 0 \Rightarrow m = -\frac{1}{2}, -\frac{1}{3}$ | M1 | Auxiliary equation |
|---|---|---|
| $x = Ae^{-\frac{1}{2}t} + Be^{-\frac{1}{3}t}$ | A1 | Complementary function. Allow with '$x=$' missing |
| $x = p + qt + rt^2 \Rightarrow x' = q + 2rt \Rightarrow x'' = 2r$ | B1 | Particular integral and its derivatives |
| $12r + 5q + 10rt + p + qt + rt^2 = t^2 + 10t + 13$ | M1 | Substitutes and equates coefficients |
| $r = 1,\quad q = 0,\quad p = 1$ | A1 | |
| $x = Ae^{-\frac{1}{2}t} + Be^{-\frac{1}{3}t} + t^2 + 1$ | A1 | Must have $x=$ |
## Question 2(b):
| $x = t^2 + 1$ | B1FT | Must have '$x=$'. Accept '$x$ is approximately' but do not accept '$x \rightarrow$'. FT on their PI |
---2 The variables $x$ and $y$ are related by the differential equation
$$6 \frac { d ^ { 2 } x } { d t ^ { 2 } } + 5 \frac { d x } { d t } + x = t ^ { 2 } + 10 t + 13$$
\begin{enumerate}[label=(\alph*)]
\item Find the general solution for $x$ in terms of $t$.
\item State an approximate solution for large positive values of $t$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2023 Q2 [7]}}