General Solution of Riccati Equation from Particular Solution

Theorem
Consider the Riccati equation:
 * $(1): \quad y' = \map p x + \map q x y + \map r x y^2$

Let $\map {y_1} x$ be a particular solution to $(1)$.

Then the general solution to $(1)$ has the form:
 * $\map y x = \map {y_1} x + \map z x$

where $\map z x$ is the general solution to the Bernoulli equation:
 * $z' - \paren {q - 2 r y_1} z = r z^2$

Proof
Let $\map y x = \map {y_1} x + \map z x$ be a particular solution to $(1)$.

Then: