I have this ODE:
$$p = xp' - x(p^2 + x^2)$$
after dividing by $x \neq 0$ we get
$$ p' = \frac{p}{x} + p^2 + x^2 $$
Which I recognized as a Riccati equation, and also confirmed this on WolframAlpha.
However, here's the problem. We always dealt with three subtypes of the Riccati equation:
a) For any Riccati equation $y' = P(x)y^2 + Q(x)y + R(x)$, if P(x), Q(x) and R(x) are constants, then it is a separable equation
b) If $y' = Ay^2 + \frac{B}{x}y + \frac{C}{x^2}$, A,B,C are constants, we let $z=yx, z=z(x)$
c) If a particular solution $y_1$ is known, we let $y(x) = y_1(x) + \frac{1}{z(x)}$
However, my equation does not fall into any of the three categories I mentioned and I don't know how to proceed. Any help would be appreciated!