Classification of initial data for the Riccati equation

Chernyavskaya, N. ; Shuster, L.

Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 511-525

Résumé

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 $q(x)$.