2017 AJC P1 Q5
A curve $C$ has equation $y=\text{f}\left( x \right)$. 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 $y=\text{f}\left( x \right)$satisfies the equation $\left( 1+2x \right)\frac{{{\text{d}}^{2}}y}{{{\text{d}}^{2}}x}+y\frac{\text{d}y}{\text{d}x}=0$ and that the third term in the Maclaurin’s expansion of $\text{f}\left( x \right)$is $\frac{1}{3}{{x}^{2}}$.
Find the value of $a$. Hence, find the Maclaurin’s series for $\text{f}\left( x \right)$in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.
[7]
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