Explanation

(a)
The given equation is
$t = 1 + (ty)\left( \frac{dy}{dt} \right) + \frac{(ty)^{2}}{2!}\left( \ \frac{dy}{dt} \right)^{2} + ...\infty$
$\Rightarrow t = e^{ty\left( \frac{dy}{dt} \right)}$
$\Rightarrow \log{t = ty\ \frac{dy}{dt}}$
$\Rightarrow y\ dy = \frac{\log t}{t}dt$
On integrating both sides, we get
$\frac{y^{2}}{2} = \frac{({\log{t)}}^{2}}{2} + k$
$\Rightarrow y = \pm \sqrt{({\log{t)}}^{2} + 2k}$
$\Rightarrow y = \pm \sqrt{({\log{t)}}^{2} + c}$