CAIE A-Level Further Math A2 2.6.1 Linear Differential Equations Questions

Practise solving first-order linear differential equations with integrating factors and initial conditions using Further Mathematics Paper 2 questions and mark schemes.

Exam points

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

[Maximum number: 4]

Show that an appropriate integrating factor for

\left(x^{2}+1\right) \frac{\mathrm{d} y}{\mathrm{~d} x}+y \sqrt{x^{2}+1}=x^{2}-x \sqrt{x^{2}+1}

is $x+\sqrt{x^{2}+1}$ .

$\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{y}{\sqrt{ }\left(x^{2}+1\right)}=\frac{x}{x^{2}+1}\left(x-\sqrt{ }\left(x^{2}+1\right)\right)$ $\mathrm{e}^{\int \frac{1}{\sqrt{\left(x^{2}+1\right)}} \mathrm{d} x}=\mathrm{e}^{\sinh ^{-1} x}$ $=x+\sqrt{ }\left(x^{2}+1\right)$
7a 4

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CAIE A-Level Further Math A2 2.6.1 Linear Differential Equations Questions question 1