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.
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.