A question regarding substituting $1/x'(y)$ for $y'(x)$ in the process of solving differential equations

In the book: "Differential Equations With Applications And Historical Notes. Third Edition" by George F. Simmons, one of the exercise problems is as follows:

The usual notation $\frac{dy}{dx}$ implies that $x$ is the independent variable and $y$ is the dependent variable. In trying to solve a differential equation, it is sometimes helpful to replace $x$ by $y$ and $y$ by $x$ and work on the resulting equation. Apply this method to the following equations.

I'm slightly uncertain about what exactly he means by "swapping x and y", but given one of the equations right below:

$(e^y-2xy)*y' = y^2 \rightarrow$

$y'=\frac{y^2}{e^y-2xy}$

if we were working on this same equation, but with the right side reciprocated, then we could do as follows:

$y'=\frac{e^y}{y^2}-\frac{2xy}{y^2} \rightarrow$

$y' + \frac{y}{2x}=e^{y}*y^{-2}$

This is a bernoulli's equation, and this indeed turns out to be a good strategy for solving this ODE.

At numerous other points in the book too, actually especially in proofs and in many steps implicitely (as in not directly stated by the author) made between intermediate results, this manipulation of derivatives is used. And what else exactly would he mean by swapping x and y? Just replacing them in terms of purely the letters would only have a cosmetic effect.

The reason this manipulation confuses me is simple: we have no idea - knowing very little about the solution - that $y'(x)$ is one to one. We don't know if the function $x'(y)$ actually exists. So, concretely, my two questions are as follows:

1. Can we - and if so why - use this manipulation when dealing with differential equations?

2. If not, what exactly is the author refering to here?

• I too have this book as my coursebook and from what I understood he means that (if we disregard the definition and limits,domin,ranges,etc) if we learnt a formula wrt to x then a question could have an easier solution if the variables are swapped in order to find the type solution the equation could have...hope this helps...also bits? – Pi_die_die Jul 15 '18 at 18:29
• @Pi_die_die hmm, but in something quite rigirous as differential equations (for example the theory about the uniqueness of particular solutions which is very technical and comes from analysis) can we allow ourselves to ignore definitions, limits, domain, ranges, ect? – Buster Bie Jul 15 '18 at 18:33
• No we can't...and you will see that he doesn't...but as of that chapter he just wants to give us tools to solve a first order ode... – Pi_die_die Jul 15 '18 at 18:35

At each point where $y'(x) \neq 0$ (and assuming that $y(x)$ is continuously differentiable, which is usually automatically true if it satisfies the ODE), the implicit function theorem says that you can at least locally invert the function $y(x)$.