2017 AJC P1 Q5

A curve \(C\) has equation \(y=\mathrm{f}\left( x \right)\). The equation of the tangent to the curve \(C\) at the point where \(x=0\) is given by \(2x-ay=3\) where \(a\) is a positive constant.

It is also given that \(y=\mathrm{f}\left( x \right)\)satisfies the equation \(\left( 1+2x \right)\frac{{{\mathrm{d}}^{2}}y}{{{\mathrm{d}}^{2}}x}+y\frac{\mathrm{d}y}{\mathrm{d}x}=0\) and that the third term in the Maclaurin’s expansion of \(\mathrm{f}\left( x \right)\)is \(\frac{1}{3}{{x}^{2}}\).

Find the value of \(a\). Hence, find the Maclaurin’s series for \(\mathrm{f}\left( x \right)\)in ascending powers of \(x\), up to and including the term in \({{x}^{3}}\).[7]