| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Standard +0.3 Part (a) is straightforward verification requiring differentiation of a given function and substitution - routine calculus with no problem-solving. Part (b) requires finding the complementary function (repeated root case), recognizing the given particular integral, applying initial conditions, and combining solutions. While this involves multiple standard techniques from FP2, it follows a well-established procedure without requiring novel insight or particularly challenging algebraic manipulation. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
\begin{enumerate}[label=(\alph*)]
\item Show that $y = \frac{1}{2}x^2e^x$ is a solution of the differential equation
$$\frac{d^2 y}{dx^2} - 2\frac{dy}{dx} + y = e^x.$$ [4]
\item Solve the differential equation
$$\frac{d^2 y}{dx^2} - 2\frac{dy}{dx} + y = e^x,$$
given that at $x = 0$, $y = 1$ and $\frac{dy}{dx} = 2$. [9]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q3 [13]}}