Geodesic
Classification of initial data for the Riccati equation
Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 511-525.

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We consider a Cauchy problem $$y'(x)+y^{2}(x)= q(x),\qquad y(x)|_{x=x_{0}}=y_{0}$$ where $x_{0}$ , $y_{0}\in \mathbb{R}$ and $q(x)\in L_{1}^{\text{loc}}(R)$ is a non-negative function satisfying the condition: $$\int_{-\infty}^{x} q(t)\, dt> 0, \quad \int_{x}^{\infty} q(t) \, dt> 0 \qquad \text{ for } x\in \mathbb{R}.$$ We obtain the conditions under which $y(x)$ can be continued to all of $\mathbb{R}$. This depends on $x_{0}$ , $y_{0}$ and the properties of $q(x)$.
Consideriamo un problema di Cauchy $$y'(x)+y^{2}(x)= q(x),\qquad y(x)|_{x=x_{0}}=y_{0}$$ dove $x_{0}$ , $y_{0}\in \mathbb{R}$ e $q(x)\in L_{1}^{\text{loc}}(\mathbb{R})$ è una funzione non negativa che soddisfa la condizione: $$\int_{-\infty}^{x} q(t) \, dt > 0, \quad \int_{x}^{\infty} q(t) \, dt> 0 \qquad \text{ for } x\in \mathbb{R}.$$ Otteniamo le condizioni nelle quali $y(x)$ può essere continuata in tutto $\mathbb{R}$. Questo dipende da $x_{0}$, $y_{0}$ e dalle proprietà di $q(x)$. 
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     title = {Classification of initial data for the {Riccati} equation},
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Chernyavskaya, N.; Shuster, L. Classification of initial data for the Riccati equation. Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 511-525. http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_2_a12/

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