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:
\(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds y_1' + z'\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map p x + \map q x \paren {y_1 + z} + \map r x \paren {y_1 + z}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map p x + \map q x \paren {y_1 + z} + \map r x \paren {y_1^2 + 2 y_1 z + z^2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds z'\) | \(=\) | \(\ds \map q x z + \map r x \paren {2 y_1 z + z^2}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z' - \paren {\map q x + 2 y_1 \map r x z}\) | \(=\) | \(\ds \map r x z^2\) |
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): Miscellaneous Problems for Chapter $2$: Problem $28$