CAIE A-Level Further Math A2 2.6.6 Initial Conditions Questions

$m^{2}+2 m+3=0$
Auxiliary equation.

$[y=] \mathrm{e}^{-x}(A \cos \sqrt{2} x+B \sin \sqrt{2} x)$
Complementary function. Allow with " y= " missing.

$y=p x^{2}+q x+r \Rightarrow y^{\prime}=2 p x+q \Rightarrow y^{\prime \prime}=2 p$
Particular integral and its derivatives.

$3 p=27 \quad 3 q+4 p=0 \quad 2 p+2 q+3 r=0$
Substitutes and equates coefficients.

4

$p=9 \quad q=-12 \quad r=2$$y=\mathrm{e}^{-x}(A \cos \sqrt{2} x+B \sin \sqrt{2} x)+9 x^{2}-12 x+2$
General solution. Must have " y= ".

$y^{\prime}=\mathrm{e}^{-x}(-\sqrt{2} A \sin \sqrt{2} x+\sqrt{2} B \cos \sqrt{2} x)-\mathrm{e}^{-x}(A \cos \sqrt{2} x+B \sin \sqrt{2} x)+18 x-12$

M1*

Differentiates. Must use product rule.

$$\begin{aligned} A+2=2 \quad \sqrt{2} B-A-12=-8 \Rightarrow A=0, B=2 \sqrt{2} \end{aligned}$$
Uses initial conditions

$y=2 \sqrt{2} \mathrm{e}^{-x} \sin \sqrt{2} x+9 x^{2}-12 x+2$
Must have " y= ".

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