Question 8 - A-Level Maths

CAIE FP1 2009 June — Question 8 8 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2009
Session	June
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Second order differential equations
Type	Asymptotic behavior for large values
Difficulty	Challenging +1.2 This is a standard second-order linear ODE with constant coefficients requiring auxiliary equation solution (complex roots), particular integral by inspection/undetermined coefficients (polynomial form), then asymptotic analysis. While it involves multiple techniques and the asymptotic behavior part requires understanding dominant terms, it follows a well-established procedure taught in Further Maths with no novel insight required. The complex roots and final limit analysis elevate it slightly above average difficulty.
Spec	4.10e Second order non-homogeneous: complementary + particular integral

8 Find the general solution of the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 65 y = 65 x ^ { 2 } + 8 x + 73$$ Show that, whatever the initial conditions, $\frac { y } { x ^ { 2 } } \rightarrow 1$ as $x \rightarrow \infty$.

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Answer	Marks
Roots of AQE are $-1/2 + 4i$ (allow 1 sign/coefficient error)	M1
$CF = e^{-x/2}(A\sin 4x + B\cos 4x)$	A1
PI of the form $ax^2 + bx + c$	M1
$PI = x^2 + 1$	M1A1
General solution: $y = e^{-x/2}(A\sin 4x + B\cos 4x) + x^2 + 1$	A1
$CF/x^2 \to 0 \forall A$ and $B \Rightarrow CF/x^2 \to 0$	M1
$\Rightarrow y/x^2 \to 1$ as $x \to \infty$ (AG, CWO)	A1
Allow $e^{-Kx} \to 0$ as $x \to \infty$ provided $K > 0$ for M1
[Use of D operator for PI	M1
Obtaining PI	M1A1]

Roots of AQE are $-1/2 + 4i$ (allow 1 sign/coefficient error) | M1 |

$CF = e^{-x/2}(A\sin 4x + B\cos 4x)$ | A1 |

PI of the form $ax^2 + bx + c$ | M1 |

$PI = x^2 + 1$ | M1A1 |

General solution: $y = e^{-x/2}(A\sin 4x + B\cos 4x) + x^2 + 1$ | A1 |

$CF/x^2 \to 0 \forall A$ and $B \Rightarrow CF/x^2 \to 0$ | M1 |

$\Rightarrow y/x^2 \to 1$ as $x \to \infty$ (AG, CWO) | A1 |

Allow $e^{-Kx} \to 0$ as $x \to \infty$ provided $K > 0$ for M1 |

[Use of D operator for PI | M1 |

Obtaining PI | M1A1] |

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8 Find the general solution of the differential equation

$$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 65 y = 65 x ^ { 2 } + 8 x + 73$$

Show that, whatever the initial conditions, $\frac { y } { x ^ { 2 } } \rightarrow 1$ as $x \rightarrow \infty$.

\hfill \mbox{\textit{CAIE FP1 2009 Q8 [8]}}

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Q1 5 Q2 6 Q3 6 Q4 6 Q5 7 Q6 7 Q7 8 Q8 8 Q9 11 Q10 11 Q11 12 Q12 EITHER Q12 OR

Answer	Marks
Roots of AQE are \(-1/2 + 4i\) (allow 1 sign/coefficient error)	M1
\(CF = e^{-x/2}(A\sin 4x + B\cos 4x)\)	A1
PI of the form \(ax^2 + bx + c\)	M1
\(PI = x^2 + 1\)	M1A1
General solution: \(y = e^{-x/2}(A\sin 4x + B\cos 4x) + x^2 + 1\)	A1
\(CF/x^2 \to 0 \forall A\) and \(B \Rightarrow CF/x^2 \to 0\)	M1
\(\Rightarrow y/x^2 \to 1\) as \(x \to \infty\) (AG, CWO)	A1
Allow \(e^{-Kx} \to 0\) as \(x \to \infty\) provided \(K > 0\) for M1
[Use of D operator for PI	M1
Obtaining PI	M1A1]