TITLE: particular solution guessing of a Riccati equation
QUESTION [5 upvotes]: I can construct the general solution of a Riccati equation if a particular solution is known. Is there any guessing trick do exist for example:
  $$\text{For }y'+y^2=\frac{2}{x^2} \text{ seek for a P.S. in the form }y=\frac{c}{x}$$
$$\text{For }x^3y'+x^2y-y^2=2x^4 \text{ seek for a P.S. in the form }y=cx^2$$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\vdots$

Why seeking for a particular solution$(\text{P.S.})$ in that form$?$ And how the knew that will be work$?$ I really appreciated any kind of help.

REPLY [2 votes]: $$y'+y^2=\frac{2}{x^2} $$
You can try a solution like $y=ax^m$ then:
$$amx^{m-1}+a^2x^{2m}=2x^{-2}$$
We need to have all the eponents be equals so
$$m-1=2m \implies m=-1$$
You have that:
$$-a+a^2=2 \implies a=-1,a=2$$
It's the same trick for the second equation try $ax^m$
These equations you are given are really particular ones. And they are hard to solve if you only change the exponent on the right side.