Taylor Series (Edexcel A Level Further Maths: Further Pure 1): Exam Questions

The Taylor series expansion of $\mathrm{f}\left(x\right)$ about $x=a$ is given by

$\mathrm{f}\left(x\right)=\mathrm{f}\left(a\right)+\mathrm{f}\text{'}\left(a\right)(x-a)+\frac{\mathrm{f}\text{'}\text{'}\left(a\right)}{2!}(x-a{)}^2+\dots +\frac{{\mathrm{f}}^{\left(r\right)}\left(a\right)}{r!}(x-a{)}^r+\dots$

The curve with equation $y=\mathrm{f}\left(x\right)$ satisfies the differential equation

$\frac{{\text{d}}^{2}y}{\text{d}{x}^2}+y\frac{\text{d}y}{\text{d}x}-y\text{sin}x=0$

Given that $\left(\frac{\pi }{6},2\right)$ is a stationary point of the curve,

determine the nature of this stationary point, giving a reason for your answer.

Show that $\frac{{\text{d}}^{3}y}{\text{d}{x}^3}=\sqrt{3}-2$ at this stationary point.

Hence determine a series solution for $y$, in ascending powers of $\left(x-\frac{\pi }{6}\right)$ up to and including the term in ${\left(x-\frac{\pi }{6}\right)}^3$, giving each coefficient in simplest form.