| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Verify particular integral form |
| Difficulty | Standard +0.3 This is a standard Further Maths second-order differential equation question requiring substitution of a given particular integral form to find k, then combining with the complementary function. The particular integral form is provided (not requiring students to determine it), and the repeated root case (auxiliary equation (m-5)²=0) is straightforward. Slightly above average difficulty due to being Further Maths content and requiring careful differentiation of the product x²e^(5x), but otherwise routine application of standard techniques. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y_{PI} = kx^2e^{5x} \Rightarrow y' = 2kxe^{5x} + 5kx^2e^{5x}\) | M1, A1 | Product rule to differentiate \(x^2e^{5x}\) |
| \(\Rightarrow y'' = 2ke^{5x} + 10kxe^{5x} + 10kxe^{5x} + 25kx^2e^{5x}\) | A1ft | |
| \(\Rightarrow 2ke^{5x} + 20kxe^{5x} + 25kx^2e^{5x}\) | ||
| \(-10(2kxe^{5x} + 5kx^2e^{5x}) + 25kx^2e^{5x} = 6e^{5x}\) | M1, A1 | Substitution into differential equation |
| \(2k = 6 \Rightarrow k = 3\) | A1ft | 6 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Aux. eqn. \(m^2 - 10m + 25 = 0 \Rightarrow m = 5\) | B1 | PI |
| CF is \((A + Bx)e^{5x}\) | M1 | |
| GS \(y = (A + Bx)e^{5x} + 3x^2e^{5x}\) | M1, A1ft | 4 marks |
## Question 1:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y_{PI} = kx^2e^{5x} \Rightarrow y' = 2kxe^{5x} + 5kx^2e^{5x}$ | M1, A1 | Product rule to differentiate $x^2e^{5x}$ |
| $\Rightarrow y'' = 2ke^{5x} + 10kxe^{5x} + 10kxe^{5x} + 25kx^2e^{5x}$ | A1ft | |
| $\Rightarrow 2ke^{5x} + 20kxe^{5x} + 25kx^2e^{5x}$ | | |
| $-10(2kxe^{5x} + 5kx^2e^{5x}) + 25kx^2e^{5x} = 6e^{5x}$ | M1, A1 | Substitution into differential equation |
| $2k = 6 \Rightarrow k = 3$ | A1ft | **6 marks** | Only ft if $xe^{5x}$ and $x^2e^{5x}$ terms all cancel out |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Aux. eqn. $m^2 - 10m + 25 = 0 \Rightarrow m = 5$ | B1 | PI |
| CF is $(A + Bx)e^{5x}$ | M1 | |
| GS $y = (A + Bx)e^{5x} + 3x^2e^{5x}$ | M1, A1ft | **4 marks** | Their CF + their/our PI; ft only on wrong value of $k$ |
---1
\begin{enumerate}[label=(\alph*)]
\item Find the value of the constant $k$ for which $k x ^ { 2 } \mathrm { e } ^ { 5 x }$ is a particular integral of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 10 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 25 y = 6 \mathrm { e } ^ { 5 x }$$
\item Hence find the general solution of this differential equation.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2007 Q1 [10]}}