The differential equation $$\[ \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+6 \frac{\mathrm{d} x}{\mathrm{~d} t}+13 x=8 \mathrm{e}^{-3 t} \quad t \geqslant 0 \]$$ describes the motion of a particle along the $$\(x\)$$-axis. Given that the motion of the particle satisfies $$\(x=\frac{1}{2}\)$$ and $$\(\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{1}{2}\)$$ when $$\(t=0\)$$ (b) determine the particular solution for the motion of the particle. (4)

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