Second order reaction cases https://chem.libretexts.org/Textbook_Maps/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Kinetics/Reaction_Rates/Second-Order_Reactions

When you will read this webpage ,then you will find that they have made cases for different types of second order reaction .

In  $situation 2A $ of case 2 in  the article (eq 1.13 and 1.14) ,they have taken change in concentration of A to be dx/dt instead of
d(a-x)/dt .What is the reason for this ?

Also what I am also finding difficulty is how did they combine the reactants to make it $[A]^2$.......  it would mean that individually the concentration of a single reactant A  is following 1st order reaction. So the rate constant in disappearence of A should be first order instead of second so what is the reason for this .I t would have make sense for 2A to follow second order instead of just A
If you are talking about the 1.13 and 1.14 equations, than whta he did was just a simple subistituction where $[A] = x,\ d[A] = dx$, if not, please identify the step.

And, yes, you are going to get diferent equations for $[A]_{(t)}$ depending on the order of reaction, $0^{th}$ order gives you a linear equation, $1^{st}$ order gives you an exponetial equation, and all the orders higher than 1 gives you a fraction diference like: $$\frac{1}{A_t^n}-\frac{1}{A_0^n} = n\ k\ t$$ where n is the order of reaction minus 2.

The step where they make $[A]^2$ is only because you have an elementary equation, it can not be assumed to be that if the equation was golbal or not elementary.