$$ \frac{d^{2}y}{dx^2} +a_{1}\left(x\right)\frac{dy}{dx} +a_{0}\left(x\right) y=F\left(x\right) \tag{1} $$
$$a_{0}\left(x\right),a_{1}\left(x\right),F\left(x\right)~ \text{are predefined} $$
$$ t:= \xi\left(x\right) \tag{2} $$
$$ \frac{dy}{dx} = \frac{dy}{dt} \frac{dt}{dx}= \frac{dy}{dt}\xi'\left(x\right)\tag{3} $$
$$ \frac{d^{2}y}{dx^2}= \frac{d}{dx}\left(\frac{dy}{dx} \right) \tag{4} $$
$$ = \frac{d}{dx}\left(\frac{dy}{dt}\xi'\left(x\right)\right) $$
$$ = \xi''\left(x\right)\frac{dy}{dt}+ \underbrace{\xi'\left(x\right)\frac{d}{dx}\left(\frac{dy}{dt}\right)}_{=:A} $$
$$ = \xi''\left(x\right)\frac{dy}{dt}+ \underbrace{\left( \xi'\left(x\right) \right)^{2} \frac{d^{2}y}{dt^2}}_{\text{how this can be derived?} } $$
$$ A= \xi'\left(x\right)\frac{d}{dx}\left(\frac{dy}{dt}\right) $$
$$ = \xi'\left(x\right)\frac{d}{dx}\left( \frac{dy}{dx} \frac{dx}{dt}\right)~~ \leftarrow~~ \text{my hand stopped from here} $$
Denote for clarity $\frac{dy}{dt}=v(t)$. Then $$ \frac{d}{dx}\left( \frac{dy}{dt}\right)=\frac{dv(t(x))}{dx} =\frac{dv}{dt}\cdot\frac{dt}{dx}=\frac{d^{2}y}{dt^2}\cdot\xi'(x). $$