CAIE A-Level Further Math A2 2.6.4 Linear Differential Equations Questions

Practise finding general solutions of higher-order linear differential equations with Further Mathematics Paper 2 questions and mark schemes.

Exam points

Apply de Moivre's theorem to powers and roots of complex numbers.

[Maximum number: 7]

It is given that $x=t^{3} y$ and

t^{3} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} t^{2}}+\left(4 t^{3}+6 t^{2}\right) \frac{\mathrm{d} y}{\mathrm{~d} t}+\left(13 t^{3}+12 t^{2}+6 t\right) y=61 \mathrm{e}^{\frac{1}{2} t} .

Find the general solution for y in terms of t.

$m^{2}+4 m+13=0 \Rightarrow m=-2 \pm 3 i$ $x=\mathrm{e}^{-2 t}(A \cos 3 t+B \sin 3 t)$ $x=k \mathrm{e}^{\frac{1}{2} t} \Rightarrow \dot{x}=\frac{1}{2} k \mathrm{e}^{\frac{1}{2} t} \Rightarrow \ddot{x}=\frac{1}{4} k \mathrm{e}^{\frac{1}{2} t}$ $\frac{1}{4} k+2 k+13 k=61 \Rightarrow k=4$ $t^{3} y=\mathrm{e}^{-2 t}(A \cos 3 t+B \sin 3 t)+4 \mathrm{e}^{\frac{1}{2} t} \Rightarrow y=t^{-3} \mathrm{e}^{-2 t}(A \cos 3 t+B \sin 3 t)+4 t^{-3} \mathrm{e}^{\frac{1}{2} t}$
7b 7

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CAIE A-Level Further Math A2 2.6.4 Linear Differential Equations Questions question 1