2009 TPJC P1 Q11

Given that \(y={{\mathrm{e}}^{{{\tan }^{-1}}x}}\), find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) and show that \(\left( 1+{{x}^{2}} \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+\left( 2x-1 \right)\frac{\mathrm{d}y}{\mathrm{d}x}=0\). Obtain the Maclaurin series for \({{\mathrm{e}}^{{{\tan }^{-1}}x}}\) up to and including the term in \({{x}^{2}}\).[5]

Expand \({{\left( 1+3x \right)}^{-\frac{1}{2}}}\) in ascending powers of \(x\) up to and including the term in \({{x}^{2}}\).[2]

Using the series that you obtained in (i) and (ii), deduce the equation of the tangent to the curve \(y=\frac{{{\mathrm{e}}^{{{\tan }^{-1}}x}}}{\sqrt{1+3x}}\) at \(x=0\).[2]