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If \( y=e^{a x} \cos b x \), prove that \( y_{2}-2 a y_{1}+\left(a^{2}+b^{2}\right) y=0 \) and hence that \( e^{a x} \cos b x=1+a x+\frac{a^{2}-b^{2}}{2 !} x^{2}+\frac{a\left(a^{2}-3 b^{2}\right)}{3 !} x^{3}+\ldots \)
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