| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Challenging +1.2 This is a standard Further Maths second-order linear differential equation with constant coefficients and a sinusoidal forcing term. While it requires multiple techniques (auxiliary equation, particular integral by trial solution, applying initial conditions), these are all routine procedures taught systematically in F2. The question is harder than typical A-level maths due to being Further Maths content, but it's a textbook-standard example within that syllabus with no novel problem-solving required. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| AE: \(m^2 - 2m - 3 = 0\) | — | — |
| \(m^2 - 2m - 3 = 0 \Rightarrow m = -1, 3\) | M1 | Forms Auxiliary Equation and attempts to solve |
| \((y=)\ Ae^{3x} + Be^{-x}\) | A1 | Cao |
| PI: \((y=)\ p\sin x + q\cos x\) | B1 | Correct form for PI |
| \((y'=)\ p\cos x - q\sin x\) | — | — |
| \((y''=)\ -p\sin x - q\cos x\) | — | — |
| \(-p\sin x - q\cos x - 2(p\cos x - q\sin x) - 3p\sin x - 3q\cos x = 2\sin x\) | M1 | Differentiates twice and substitutes |
| \(2q - 4p = 2,\quad 4q + 2p = 0\) | A1 | Correct equations |
| \(p = -\frac{2}{5},\quad q = \frac{1}{5}\) | A1A1 | A1A1 both correct; A1A0 one correct |
| \(y = Ae^{3x} + Be^{-x} + \frac{1}{5}\cos x - \frac{2}{5}\sin x\) | B1ft | Follow through their \(p\) and \(q\) and their CF |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y' = 3Ae^{3x} - Be^{-x} - \frac{1}{5}\sin x - \frac{2}{5}\cos x\) | M1 | Differentiates their GS |
| \(0 = A + B + \frac{1}{5},\quad 1 = 3A - B - \frac{2}{5}\) | M1A1 | M1: Uses given conditions to give two equations in \(A\) and \(B\); A1: Correct equations |
| \(A = \frac{3}{10},\quad B = -\frac{1}{2}\) | A1 | Solves for \(A\) and \(B\), both correct |
| \(y = \frac{3}{10}e^{3x} - \frac{1}{2}e^{-x} + \frac{1}{5}\cos x - \frac{2}{5}\sin x\) | A1ft | Sub their values of \(A\) and \(B\) into their GS |
## Question 6:
**Part (a):**
| Answer/Working | Marks | Guidance |
|---|---|---|
| AE: $m^2 - 2m - 3 = 0$ | — | — |
| $m^2 - 2m - 3 = 0 \Rightarrow m = -1, 3$ | M1 | Forms Auxiliary Equation and attempts to solve |
| $(y=)\ Ae^{3x} + Be^{-x}$ | A1 | Cao |
| PI: $(y=)\ p\sin x + q\cos x$ | B1 | Correct form for PI |
| $(y'=)\ p\cos x - q\sin x$ | — | — |
| $(y''=)\ -p\sin x - q\cos x$ | — | — |
| $-p\sin x - q\cos x - 2(p\cos x - q\sin x) - 3p\sin x - 3q\cos x = 2\sin x$ | M1 | Differentiates twice and substitutes |
| $2q - 4p = 2,\quad 4q + 2p = 0$ | A1 | Correct equations |
| $p = -\frac{2}{5},\quad q = \frac{1}{5}$ | A1A1 | A1A1 both correct; A1A0 one correct |
| $y = Ae^{3x} + Be^{-x} + \frac{1}{5}\cos x - \frac{2}{5}\sin x$ | B1ft | Follow through their $p$ and $q$ and their CF |
**Total: (8)**
---
**Part (b):**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y' = 3Ae^{3x} - Be^{-x} - \frac{1}{5}\sin x - \frac{2}{5}\cos x$ | M1 | Differentiates their GS |
| $0 = A + B + \frac{1}{5},\quad 1 = 3A - B - \frac{2}{5}$ | M1A1 | M1: Uses given conditions to give two equations in $A$ and $B$; A1: Correct equations |
| $A = \frac{3}{10},\quad B = -\frac{1}{2}$ | A1 | Solves for $A$ and $B$, both correct |
| $y = \frac{3}{10}e^{3x} - \frac{1}{2}e^{-x} + \frac{1}{5}\cos x - \frac{2}{5}\sin x$ | A1ft | Sub their values of $A$ and $B$ into their GS |
**Total: (5)**
---\begin{enumerate}
\item (a) Find the general solution of the differential equation
\end{enumerate}
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = 2 \sin x$$
Given that $y = 0$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1$ when $x = 0$\\
(b) find the particular solution of differential equation (I).\\
\hfill \mbox{\textit{Edexcel F2 2015 Q6 [13]}}