| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Solve via substitution then back-substitute |
| Difficulty | Challenging +1.3 This is a structured FP3 differential equations question where part (a) guides students through the substitution with a given result to verify, and part (b) requires solving a standard second-order linear DE with constant coefficients (d²u/dx² + u = sin 2x). The substitution involves careful but routine differentiation using the product rule, and the final DE is a textbook example requiring complementary function plus particular integral. While requiring multiple techniques and careful algebra, the scaffolding in part (a) and the standard nature of the resulting equation place this slightly above average difficulty but well within expected FP3 scope. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
7 A differential equation is given by
$$\sin ^ { 2 } x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \sin x \cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 2 \sin ^ { 4 } x \cos x , \quad 0 < x < \pi$$
\begin{enumerate}[label=(\alph*)]
\item Show that the substitution
$$y = u \sin x$$
where $u$ is a function of $x$, transforms this differential equation into
$$\frac { \mathrm { d } ^ { 2 } u } { \mathrm {~d} x ^ { 2 } } + u = \sin 2 x$$
\item Hence find the general solution of the differential equation
$$\sin ^ { 2 } x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \sin x \cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 2 \sin ^ { 4 } x \cos x$$
giving your answer in the form $y = \mathrm { f } ( x )$.\\
(6 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2013 Q7 [11]}}