| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Session | Specimen |
| Marks | 9 |
| Topic | Second order differential equations |
| Type | Standard non-homogeneous with polynomial RHS |
| Difficulty | Standard +0.8 This is a second-order non-homogeneous differential equation requiring: (1) solving the auxiliary equation with complex roots, (2) finding a particular integral with polynomial trial solution requiring coefficient matching, and (3) proving an asymptotic limit result. The multi-step nature, complex roots, and the non-routine part (ii) requiring analysis of limiting behavior elevate this above standard A-level questions. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
**(i)** Roots of AQE are $-\frac{1}{2} \pm 4i$ M1A1
CF $= e^{-\frac{1}{2}x}(A\sin 4x + B\cos 4x)$ A1
PI of the form $ax^2 + bx + c$ M1
PI $= x^2 + 1$ M1A1
General solution $y = e^{-\frac{1}{2}x}(A\sin 4x + B\cos 4x) + x^2 + 1$ A1 **[7]**
**(ii)** $e^{-\frac{1}{2}x} \to 0$ and $\frac{1}{x^2} \to 0$ as $x \to \infty$ M1
$\Rightarrow \frac{y}{x^2} \to 1$ as $x \to \infty$ (CWO) A1 **[2]**6 (i) Find the general solution of the differential equation
$$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 65 y = 65 x ^ { 2 } + 8 x + 73 .$$
(ii) Show that, whatever the initial conditions, $\frac { y } { x ^ { 2 } } \rightarrow 1$ as $x \rightarrow \infty$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 Q6 [9]}}