Motivation and Derivation of the Riccati Equation Transformation Given a Riccati Equation which is differential equation of the form:
$$ \frac{dy}{dx} = a_0 (x) + a_1 (x)y + a_2 (x)y^2  $$
It is well known that the transformation:
$$ y = -\frac{1}{a_2(x)} \frac{\frac{du}{dx}}{u} $$
Can be substituted into this equation to transform it into a second order linear differential equation (which after simplifying the algebra is)
$$ a_0(x)a_2(x) u - (a_1(x)a_2(x) + 1) \frac{du}{dx} + a_2(x)\frac{d^2u}{du^2} = 0 $$ 
My question is: How does one derive this transformation from scratch. Since although it is easy to prove that the transformation works it does not reveal how it was found.
 A: The important point is not that the transformed equation is of second order, but that it is linear. Often, a linear second order ODE is easier to solve than a non linear first order ODE.
If a particular solution of the Riccati equation is known, there is no interest to transform the Riccati equation, because it could be directly solved. But it is not always possible to first find a particular solution. Then, the transformation to a linear second order ODE can be helpfull in many casses, especially when the solutions involve special functions.
For example : $y' = -2\frac{y}{x}+\frac{y^2}{x}+x $ can be transformed to a second order ODE of  Bessel's kind which allows to solve the Riccati equation in terms of Bessel functions. Without this transformation, it would be almost impossble to first find a particular solution an so, quite impossible to solve the non linear first order ODE.
A: The things look more natural if you start with second order ODE $$(rx')'+cx=0$$ and use the transformation $w=\frac{rx'}{x}$ to get Riccati type equation $$w'+c+r^{-1}w^2=0.$$
The numerator of $w$ is just the quasiderivative $rx'$ and the reason for $x$ in denominator could be the homogeneity. Not sure who was the first, but a lot of info can be found in this classical book by Swanson: Comparison and oscillation theory of linear differential equations.
