| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | Specimen |
| 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 differential equation with constant coefficients requiring both complementary function (solving auxiliary equation with distinct real roots) and particular integral (trying y = A sin x + B cos x). While it involves multiple steps and differentiation of the trial solution, it follows a completely algorithmic procedure taught in F2. The initial conditions add one more routine step. It's harder than average A-level due to being Further Maths content, but it's a textbook example within that syllabus. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| AE: \(m^2 - 2m - 3 = 0 \Rightarrow m = -1, 3\) | M1 | Forms Auxiliary Equation and attempts to solve (usual rules) |
| \(y = Ae^{3x} + Be^{-x}\) | A1 | Cao |
| PI: \(y = p\sin x + q\cos x\) | B1 | Correct form for PI |
| \(y' = p\cos x - q\sin x\), \(y'' = -p\sin x - q\cos x\) | ||
| \(-p\sin x - q\cos x - 2(p\cos x - q\sin x) - 3p\sin x - 3q\cos x = 2\sin x\); differentiates twice and substitutes | M1 | |
| \(2q - 4p = 2\), \(4q + 2p = 0\) | A1 | Correct equations |
| \(p = -\frac{2}{5}\), \(q = \frac{1}{5}\) | A1 A1 | 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 |
| (8) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(y' = 3Ae^{3x} - Be^{-x} - \frac{1}{5}\sin x - \frac{2}{5}\cos x\) | M1 | Differentiates their GS |
| \(0 = A + B + \frac{1}{5}\), \(1 = 3A - B - \frac{2}{5}\) | M1 A1 | M1: Uses given conditions to give two equations in \(A\) and \(B\); A1: Correct equations |
| \(A = \frac{3}{10}\), \(B = -\frac{1}{2}\) | 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 \(B\) in their GS |
| (5) |
## Question 6(a):
$\frac{d^2y}{dx^2} - 2\frac{dy}{dx} - 3y = 2\sin x$
| Working/Answer | Mark | Guidance |
|---|---|---|
| AE: $m^2 - 2m - 3 = 0 \Rightarrow m = -1, 3$ | M1 | Forms Auxiliary Equation and attempts to solve (usual rules) |
| $y = Ae^{3x} + Be^{-x}$ | A1 | Cao |
| PI: $y = p\sin x + q\cos x$ | B1 | Correct form for PI |
| $y' = p\cos x - q\sin x$, $y'' = -p\sin x - q\cos x$ | | |
| $-p\sin x - q\cos x - 2(p\cos x - q\sin x) - 3p\sin x - 3q\cos x = 2\sin x$; differentiates twice and substitutes | M1 | |
| $2q - 4p = 2$, $4q + 2p = 0$ | A1 | Correct equations |
| $p = -\frac{2}{5}$, $q = \frac{1}{5}$ | A1 A1 | 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 |
| | **(8)** | |
## Question 6(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $y' = 3Ae^{3x} - Be^{-x} - \frac{1}{5}\sin x - \frac{2}{5}\cos x$ | M1 | Differentiates their GS |
| $0 = A + B + \frac{1}{5}$, $1 = 3A - B - \frac{2}{5}$ | M1 A1 | M1: Uses given conditions to give two equations in $A$ and $B$; A1: Correct equations |
| $A = \frac{3}{10}$, $B = -\frac{1}{2}$ | 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 $B$ in their GS |
| | **(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).
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\hfill \mbox{\textit{Edexcel F2 2018 Q6 [13]}}