| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Standard +0.8 This is a standard Further Maths second-order linear ODE with constant coefficients and polynomial forcing term. Part (a) requires finding complementary function (solving auxiliary equation with real roots) and particular integral (trying quadratic form), which is methodical but involves several algebraic steps. Part (b) applies initial conditions to find constants. While systematic, it's above average difficulty due to being Further Maths content with substantial algebraic manipulation across 14 marks. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(m^2+3m+2=0 \Rightarrow m=-1,-2\) | B1 | Correct roots (may be implied by CF) |
| \(y = Ae^{-2x}+Be^{-x}\) | M1A1 | M1: CF of correct form. A1: Correct CF |
| \(y = ax^2+bx+c\) | B1 | Correct form for PI |
| \(\frac{dy}{dx}=2ax+b,\ \frac{d^2y}{dx^2}=2a \Rightarrow 2a+3(2ax+b)+2(ax^2+bx+c)=3x^2+2x+1\) | M1 | Differentiates twice, substitutes into LHS and equates to \(3x^2+2x+1\); at least one of \(y\), \(y'\), \(y''\) correctly placed |
| \(a=\frac{3}{2}\) | A1 | — |
| \(6a+2b=2 \Rightarrow b=-\frac{7}{2},\quad c=\frac{17}{4}\) | M1A1 | M1: Solves to obtain one of \(b\) or \(c\). A1: Correct \(b\) and \(c\) |
| \(y = Ae^{-2x}+Be^{-x}+\frac{3}{2}x^2-\frac{7}{2}x+\frac{17}{4}\) | B1ft | Correct ft (their CF + their PI); must be \(y=\ldots\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(0=A+B+\frac{17}{4}\) | M1 | Substitutes \(x=0\) and \(y=0\) into GS |
| \(\frac{dy}{dx}=-2Ae^{-2x}-Be^{-x}+3x-\frac{7}{2} \Rightarrow 0=-2A-B-\frac{7}{2}\) | M1 | Attempts to differentiate and substitutes \(x=0\), \(y'=0\) |
| Solves simultaneously | M1 | Solves to obtain values for \(A\) and \(B\) |
| \(A=\frac{3}{4},\quad B=-5\) | A1 | Correct values |
| \(y=\frac{3}{4}e^{-2x}-5e^{-x}+\frac{3}{2}x^2-\frac{7}{2}x+\frac{17}{4}\) | B1ft | Correct ft (their CF + their PI); must be \(y=\ldots\) |
# Question 6:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $m^2+3m+2=0 \Rightarrow m=-1,-2$ | B1 | Correct roots (may be implied by CF) |
| $y = Ae^{-2x}+Be^{-x}$ | M1A1 | M1: CF of correct form. A1: Correct CF |
| $y = ax^2+bx+c$ | B1 | Correct form for PI |
| $\frac{dy}{dx}=2ax+b,\ \frac{d^2y}{dx^2}=2a \Rightarrow 2a+3(2ax+b)+2(ax^2+bx+c)=3x^2+2x+1$ | M1 | Differentiates twice, substitutes into LHS and equates to $3x^2+2x+1$; at least one of $y$, $y'$, $y''$ correctly placed |
| $a=\frac{3}{2}$ | A1 | — |
| $6a+2b=2 \Rightarrow b=-\frac{7}{2},\quad c=\frac{17}{4}$ | M1A1 | M1: Solves to obtain one of $b$ or $c$. A1: Correct $b$ and $c$ |
| $y = Ae^{-2x}+Be^{-x}+\frac{3}{2}x^2-\frac{7}{2}x+\frac{17}{4}$ | B1ft | Correct ft (their CF + their PI); must be $y=\ldots$ |
**Total: (9)**
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $0=A+B+\frac{17}{4}$ | M1 | Substitutes $x=0$ and $y=0$ into GS |
| $\frac{dy}{dx}=-2Ae^{-2x}-Be^{-x}+3x-\frac{7}{2} \Rightarrow 0=-2A-B-\frac{7}{2}$ | M1 | Attempts to differentiate and substitutes $x=0$, $y'=0$ |
| Solves simultaneously | M1 | Solves to obtain values for $A$ and $B$ |
| $A=\frac{3}{4},\quad B=-5$ | A1 | Correct values |
| $y=\frac{3}{4}e^{-2x}-5e^{-x}+\frac{3}{2}x^2-\frac{7}{2}x+\frac{17}{4}$ | B1ft | Correct ft (their CF + their PI); must be $y=\ldots$ |
**Total: (5)**
---6. (a) Find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 3 x ^ { 2 } + 2 x + 1$$
(9)\\
(b) Find the particular solution of this differential equation for which $y = 0$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 0$\\
(5)
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\hfill \mbox{\textit{Edexcel F2 2016 Q6 [14]}}