Originally Posted by

**rebghb** Hello everyone!

I've lately came accross this "confusion" as I was solving a cauchy-euler equation that required a subtitution of $\displaystyle e^t$ for $\displaystyle x$. It followed that $\displaystyle x=e^t\Leftrightarrow t=ln|x|.$ Then $\displaystyle dt=\frac{1}{x}\,dx$ or simply $\displaystyle \frac{dt}{dx}=\frac{1}{x}$. But what about $\displaystyle \frac{dx}{dt}$? Is it $\displaystyle x$ or $\displaystyle e^t$. I'm very confused about this because whenever we have a 1st order ODE let's say $\displaystyle y'=f(x)$ we change it to $\displaystyle \frac{dy}{dx}=f(x)$ and then multiply the RHS by $\displaystyle dx$ follows that $\displaystyle dy=f(x)\,dx$ as if the latter were a denominator!

If someone clears that out for me I would be very greatful!

Thanks!!