Conceptual doubt regarding choosing the correct method for solving a differential equation of first order but not of first degree

Question

I am learning about equations of first order but not of first degree. I have come across 3 methods of solving the equations till now:

1. Equations solvable for x
2. Equations solvable for y
3. Equations in clairaut's form

As I proceed further, I find many many cases where these techniques seem to be overlapping. To illustrate, please have a look at the following example:

$y=px+ \sqrt{(1+p^2)}$

In my opinion, this equation can be solved with all the 3 methods(As we can modify the equations in various forms to suit our requirements). But the book suggests to use Clairaut's method.

I do understand that it is much easier to solve the question with Clairaut's method, but in my opinion, the 1st two methods mentioned above can also be applied.

Am I correct in saying the above statement?

Also, whenever given any equation of 1st order but not of first degree, what should be my approach when it comes to choosing between the solution methods.

Thanks a lot in advance!