General Solution of Riccati Equation from Particular Solution

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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:

\(\ds \dfrac {\d y} {\d x}\) \(=\) \(\ds y_1' + z'\)
\(\ds \) \(=\) \(\ds \map p x + \map q x \paren {y_1 + z} + \map r x \paren {y_1 + z}^2\)
\(\ds \) \(=\) \(\ds \map p x + \map q x \paren {y_1 + z} + \map r x \paren {y_1^2 + 2 y_1 z + z^2}\)
\(\ds \leadsto \ \ \) \(\ds z'\) \(=\) \(\ds \map q x z + \map r x \paren {2 y_1 z + z^2}\)
\(\ds \leadsto \ \ \) \(\ds z' - \paren {\map q x + 2 y_1 \map r x z}\) \(=\) \(\ds \map r x z^2\)

$\blacksquare$


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