| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2002 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Resonance cases requiring modified PI |
| Difficulty | Standard +0.8 This is a Further Maths FP2 question on resonance in second-order DEs, requiring recognition that the forcing term e^x matches the complementary function (repeated root), necessitating a modified particular integral of the form x²e^x. Part (a) is verification (routine differentiation), but part (b) requires finding the CF, recognizing the resonance case, applying initial conditions, and combining solutions—several non-trivial steps beyond standard A-level. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
\begin{enumerate}[label=(\alph*)]
\item Show that $y = \frac { 1 } { 2 } x ^ { 2 } \mathrm { e } ^ { x }$ is a solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \mathrm { e } ^ { x }$$
\item Solve the differential equation $\quad \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \mathrm { e } ^ { x }$.\\
given that at $x = 0 , y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2002 Q3 [13]}}