| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2020 |
| Session | November |
| 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 differential equation with constant coefficients and polynomial RHS, requiring both complementary function (solving auxiliary equation with repeated root) and particular integral (polynomial trial solution). While the method is systematic, it involves multiple steps including handling a repeated root case and matching coefficients for a quadratic particular integral, making it moderately challenging but still routine for Further Maths students. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(9m^2 + 6m + 1 = 0 \Rightarrow m = -\frac{1}{3}\) | M1 | Auxiliary equation |
| \(y = e^{-\frac{1}{3}x}(Ax + B)\) | A1 | Complementary function |
| \(y = p + qx + rx^2 \Rightarrow y' = q + 2rx \Rightarrow y'' = 2r\) | B1 | Particular integral and its derivatives |
| \(18r + 6q + 12rx + p + qx + rx^2 = 3x^2 + 30x\) | M1 | Substitutes and equates coefficients |
| \(r = 3,\quad q = -6,\quad p = -18\) | A1 | |
| \(y = e^{-\frac{1}{3}x}(Ax + B) + 3x^2 - 6x - 18\) | A1 | |
| Total: 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = 3x^2 - 6x - 18\) | B1 FT | |
| Total: 1 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $9m^2 + 6m + 1 = 0 \Rightarrow m = -\frac{1}{3}$ | M1 | Auxiliary equation |
| $y = e^{-\frac{1}{3}x}(Ax + B)$ | A1 | Complementary function |
| $y = p + qx + rx^2 \Rightarrow y' = q + 2rx \Rightarrow y'' = 2r$ | B1 | Particular integral and its derivatives |
| $18r + 6q + 12rx + p + qx + rx^2 = 3x^2 + 30x$ | M1 | Substitutes and equates coefficients |
| $r = 3,\quad q = -6,\quad p = -18$ | A1 | |
| $y = e^{-\frac{1}{3}x}(Ax + B) + 3x^2 - 6x - 18$ | A1 | |
| **Total: 6** | | |
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = 3x^2 - 6x - 18$ | B1 FT | |
| **Total: 1** | | |2 The variables $x$ and $y$ are related by the differential equation
$$9 \frac { d ^ { 2 } y } { d x ^ { 2 } } + 6 \frac { d y } { d x } + y = 3 x ^ { 2 } + 30 x$$
\begin{enumerate}[label=(\alph*)]
\item Find the general solution for $y$ in terms of $x$.
\item State an approximate solution for large positive values of $x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2020 Q2 [7]}}