repametrization ODE My dear comrades
well an elementary question about change of parametrization in ODE :
let's say I have
$$
\frac{dx}{dt}(t) = f(t)
$$


*

*I made the change of parametrization $t\rightarrow u=s(t)$


is it correct to write 
$$
\frac{d\tilde{x}}{du}(s(t)) s'(t) = \frac{dx}{dt}(t) 
$$
ie
$$
\dot{\tilde{x}}(s(t)) = \frac{1}{s'(t)}f(t) ?
$$


*

*then the change of variables u = s(t) :


$$
\dot{\tilde{x}}(u) = \frac{1}{s'(s^{-1}(u))}f(s^{-1}(u)) ?
$$
Thanks
 A: If you make change of variables 
$\,t\to u = s\left(t\right) \,$ and denote the transformation by $\,u:\mathbb R \to \mathbb R,\,$ you might as well denote the inverse transformation $\, w:=  u^{-1}:\mathbb R \to \mathbb R,\,$ so that $\,u = s\left(t\right)\,$ and 
$\,t = w\left(u\right):\,$ 
$$
x\left(\,t\,\right) = x\,\big(\,w\left(u\right)\big) = \tilde x\left(\,u\,\right), 
\quad 
f\left(\,t\,\right) = f\,\big(\,w\left(u\right)\big) = \tilde f\left(\,u\,\right), 
$$ 
Then your differential equation will take form
$$
\frac{d\,x}{dt} =  f\left(\,t\,\right) 
\iff 
\frac{d}{dt} \big(\,x\left(\,t\,\right)\big) = f\left(\,t\,\right)
\ 
\stackrel{\substack{t &= &\,w&\,(&u&)\\u\,&=&\,s&\,(&t&)}}{\implies}
\ 
\frac{d}{dt} \Big(\,x\,\big(\,w\left(\,u\,\right)\big)\Big) 
= f\,\big(\,w\left(\,u\,\right)\big)
\iff\\\iff
\bbox[3pt, border:solid 1pt #000000]{
\frac{d}{dt} \big(\,\tilde{x}\left(\,u\,\right)\big) 
= \tilde{f}\left(\,u\,\right) }
$$
Since derivative of a function is reciprocal to  derivative of its inverse, we have 
$$
\begin{aligned}
\frac{d\,\tilde{x}}{du} &=  \frac{d\,\tilde{x}}{dt} \frac{dt}{du} 
= \frac{d\,\tilde{x}}{dt} \frac{d}{du}\big( w\left(u\right) \big) 
= \frac{d\,\tilde{x}}{dt}\, w\,'\left(u\right) 
\implies 
\frac{d\,\tilde{x}}{dt} = \frac{1}{w\,'} \frac{d\,\tilde{x}}{du} 
= s\,'\,\frac{d\,\tilde{x}}{du},
\end{aligned}
$$
and therefore
$$
\frac{d\,x}{dt} =  f\left(\,t\,\right)  
\implies 
s\,'\,\frac{d\,\tilde{x}}{du} = \tilde{f}\,\big(\,u\,\big)
\implies 
\frac{d\,\tilde{x}}{du} = \frac{1}{s\,'}\,\tilde{f}\left(\,u\,\right)
=  w\,'\left( u\right)\cdot\tilde{f}\,\big(\,u\,\big)
$$
Therefore we can rewrite it as 
$$
%\bbox[5pt, border:solid 1pt #000000]{
\frac{d\,x}{dt} =  f\left(\,t\,\right)  
\iff
\frac{d\,\tilde{x}}{dt}
= \frac{1}{s\,'}\,\tilde{f}\left(\,u\,\right)
= w\left(\,u\,\right)\cdot\tilde{f}\left(\,u\,\right)
, \quad \text{ i.e. }
\\
\bbox[5pt, border:solid 2pt #F00000]{
\frac{d\,x}{dt} =  f\left(\,t\,\right)  
\iff
\frac{d\,\tilde{x}}{du}
= \frac{\tilde{f}\,\left(\,u\,\right)}{s\,'\left(\,w\left(\,u\,\right)\,\right)}
}
$$
