| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Resonance cases requiring modified PI |
| Difficulty | Challenging +1.2 This is a standard Further Maths resonance case question with clear scaffolding. Part (i) is routine (repeated root λ=3), part (ii) tests conceptual understanding of why resonance occurs, and part (iii) involves straightforward differentiation and substitution with the given form. While it requires knowledge of modified PI for repeated roots, the structure guides students through each step, making it moderately above average difficulty but not requiring novel insight. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| \(m^2 - 6m + 9 = (0) \Rightarrow m = 3\) | M1, A1 | For attempting to solve correct auxiliary equation. For correct \(m\) |
| \(CF = (A + Bx)e^{3x}\) | A1, 3 | For correct CF |
| Answer | Marks | Guidance |
|---|---|---|
| \(ke^{3x}\) and \(kxe^{3x}\) both appear in CF | B1, 1 | For correct statement |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = kx^2e^{3x} \Rightarrow y' = 2kxe^{3x} + 3kx^2e^{3x}\) | M1, A1 | For differentiating \(kx^2e^{3x}\) twice. For correct \(y'\) aef |
| \(\Rightarrow y'' = 2ke^{3x} + 12kxe^{3x} + 9kx^2e^{3x}\) | A1 | For correct \(y''\) aef |
| \(\Rightarrow ke^{3x}\left(2 + 12x + 9y^2 - 12x - 18x^2 + 9x^2\right) = e^{3x}\) | M1 | For substituting \(y'\), \(y'\), \(y\) into DE |
| \(\Rightarrow k = \frac{1}{2}\) | A1, 5 | For correct \(k\) |
**Part (i)**
| $m^2 - 6m + 9 = (0) \Rightarrow m = 3$ | M1, A1 | For attempting to solve correct auxiliary equation. For correct $m$ |
|---|---|---|
| $CF = (A + Bx)e^{3x}$ | A1, 3 | For correct CF |
**Part (ii)**
| $ke^{3x}$ and $kxe^{3x}$ both appear in CF | B1, 1 | For correct statement |
**Part (iii)**
| $y = kx^2e^{3x} \Rightarrow y' = 2kxe^{3x} + 3kx^2e^{3x}$ | M1, A1 | For differentiating $kx^2e^{3x}$ twice. For correct $y'$ aef |
|---|---|---|
| $\Rightarrow y'' = 2ke^{3x} + 12kxe^{3x} + 9kx^2e^{3x}$ | A1 | For correct $y''$ aef |
| $\Rightarrow ke^{3x}\left(2 + 12x + 9y^2 - 12x - 18x^2 + 9x^2\right) = e^{3x}$ | M1 | For substituting $y'$, $y'$, $y$ into DE |
| $\Rightarrow k = \frac{1}{2}$ | A1, 5 | For correct $k$ |
---5 The variables $x$ and $y$ satisfy the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = \mathrm { e } ^ { 3 x }$$
(i) Find the complementary function.\\
(ii) Explain briefly why there is no particular integral of either of the forms $y = k \mathrm { e } ^ { 3 x }$ or $y = k x \mathrm { e } ^ { 3 x }$.\\
(iii) Given that there is a particular integral of the form $y = k x ^ { 2 } \mathrm { e } ^ { 3 x }$, find the value of $k$.
\hfill \mbox{\textit{OCR FP3 2009 Q5 [9]}}