| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | November |
| Marks | 14 |
| Topic | First order differential equations (integrating factor) |
| Type | Standard linear first order - constant coefficients |
| Difficulty | Standard +0.8 Part (i) requires solving a first-order linear ODE with integrating factor method and applying initial conditions—standard Further Maths content but multi-step. Part (ii) involves a second-order linear ODE with repeated roots (auxiliary equation gives (m-2)²=0) plus finding a particular integral using the given form, requiring substitution and algebraic manipulation. While the PI form is provided (reducing difficulty), the overall question demands solid technique across multiple differential equations methods, placing it moderately above average difficulty. |
| Spec | 4.10c Integrating factor: first order equations4.10e Second order non-homogeneous: complementary + particular integral |
\begin{enumerate}[label=(\roman*)]
\item Given that $y = -4$ when $x = 0$ and that
$$\frac{dy}{dx} - y = e^{2x} + 3,$$
find the value of $x$ for which $y = 0$. [7]
\item Find the general solution of
$$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = e^{2x} + 3,$$
given that $y = cx^2e^{2x} + d$ is a suitable form of particular integral. [7]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q11 [14]}}