CAIE A-Level Further Mathematics 2.6.1 Linear Differential Equations

Practise converting first-order equations to linear form and solving them with an integrating factor and initial condition.

Exam points

divide into dy/dx+P(x)y=Q(x) form before finding the integrating factor
rewrite the left side as the derivative of the integrating-factor product
integrate exactly and use the given point to determine the arbitrary constant

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

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

B1

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

M1 A1

$=x+\sqrt{ }\left(x^{2}+1\right)$

A1

7a 4

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2.6.1—Linear differential equations question 1