| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2021 |
| 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 +0.3 This is a standard second-order linear differential equation with constant coefficients and polynomial RHS. Part (a) requires routine application of auxiliary equation method (giving complementary function e^{-x} and e^{-2x}) plus particular integral (trying y=ax+b). Part (b) asks for asymptotic behavior, which simply requires recognizing exponential terms vanish. Slightly above average due to being Further Maths content, but this is a textbook example with no novel problem-solving required. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(m^2 + 3m + 2 = 0\) leading to \(m = -1,\ -2\) | M1 | Auxiliary equation. |
| \(y = Ae^{-x} + Be^{-2x}\) | A1 | Complementary function. |
| \(y = p + qx\) leading to \(y' = q\) leading to \(y'' = 0\) | B1 | Particular integral and its derivatives. |
| \(3q + 2(p + qx) = 2x + 1\) | M1 | Substitutes and equates coefficients. |
| \(q = 1 \quad p = -1\) | A1 | |
| \(y = Ae^{-x} + Be^{-2x} + x - 1\) | A1 | Must see '\(y =\)'. |
| Total: 6 |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = x - 1\) | B1 FT | Accept \(y \approx x - 1\) |
## Question 2:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $m^2 + 3m + 2 = 0$ leading to $m = -1,\ -2$ | M1 | Auxiliary equation. |
| $y = Ae^{-x} + Be^{-2x}$ | A1 | Complementary function. |
| $y = p + qx$ leading to $y' = q$ leading to $y'' = 0$ | B1 | Particular integral and its derivatives. |
| $3q + 2(p + qx) = 2x + 1$ | M1 | Substitutes and equates coefficients. |
| $q = 1 \quad p = -1$ | A1 | |
| $y = Ae^{-x} + Be^{-2x} + x - 1$ | A1 | Must see '$y =$'. |
| **Total: 6** | | |
## Question 2(b):
$y = x - 1$ | **B1 FT** | Accept $y \approx x - 1$
---2 The variables $x$ and $y$ are related by the differential equation
$$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 3 \frac { d y } { d x } + 2 y = 2 x + 1$$
\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 2021 Q2 [7]}}