| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Verify particular integral form |
| Difficulty | Moderate -0.3 This is a straightforward second-order differential equations question from Further Maths that follows a standard template: substitute the given particular integral form, equate coefficients to find constants, then add to the complementary function. While it's a Further Maths topic, the mechanical nature and clear guidance ('for which ax+b is a particular integral') make it slightly easier than average A-level difficulty overall. |
| Spec | 4.10a General/particular solutions: of differential equations4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2a - 5(ax + b) \quad (= 10x)\) | B1 | OE; PI by two correct equations or correct values for \(a\) and \(b\) |
| \(-5a = 10, \quad 2a - 5b = 0\) | M1 | Equating coefficients to form two equations, at least one correct. PI by next line |
| \(a = -2, \quad b = -\frac{4}{5}\) | A1 | Correct values for both \(a\) and \(b\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Aux eqn \(2m - 5 = 0\) | M1 | PI or solving \(2y'(x) - 5y = 0\) as far as \(y = Ae^{\pm 2.5x}\) OE |
| \((y_{CF} =) \ Ae^{2.5x}\) | A1 | OE |
| \((y_{GS} =) \ Ae^{2.5x} - 2x - 0.8\) | B1F | \((y_{GS} =)\) c's CF \(- 2x - 0.8\); must have exactly one arbitrary constant; ft c's non-zero values for \(a\) and \(b\) from part (a) |
# Question 1:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2a - 5(ax + b) \quad (= 10x)$ | B1 | OE; PI by two correct equations or correct values for $a$ and $b$ |
| $-5a = 10, \quad 2a - 5b = 0$ | M1 | Equating coefficients to form two equations, at least one correct. PI by next line |
| $a = -2, \quad b = -\frac{4}{5}$ | A1 | Correct values for both $a$ and $b$ |
**Total: 3 marks**
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Aux eqn $2m - 5 = 0$ | M1 | PI or solving $2y'(x) - 5y = 0$ as far as $y = Ae^{\pm 2.5x}$ OE |
| $(y_{CF} =) \ Ae^{2.5x}$ | A1 | OE |
| $(y_{GS} =) \ Ae^{2.5x} - 2x - 0.8$ | B1F | $(y_{GS} =)$ c's CF $- 2x - 0.8$; must have exactly one arbitrary constant; ft c's non-zero values for $a$ and $b$ from part (a) |
**Total: 3 marks**
---1
\begin{enumerate}[label=(\alph*)]
\item Find the values of the constants $a$ and $b$ for which $a x + b$ is a particular integral of the differential equation
$$2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 5 y = 10 x$$
\item Hence find the general solution of $2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 5 y = 10 x$.\\[0pt]
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2016 Q1 [6]}}