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We consider a Cauchy problem $$y^{\prime }\left(x\right)+y^2\left(x\right)=q\left(x\right),{\left.y\left(x\right)\right|}_{x=x_0}=y_0$$ where $x_0$ , $y_0\in \mathbb{R}$ and $q\left(x\right)\in L_1^{\text{loc}}\left(R\right)$ is a non-negative function satisfying the condition: $${\int }_{-\mathrm{\infty }}^{x}q\left(t\right)dt>0,{\int }_x^{\mathrm{\infty }}q\left(t\right)dt>0\text{ for }x\in \mathbb{R}.$$ We obtain the conditions under which $y\left(x\right)$ can be continued to all of $\mathbb{R}$. This depends on $x_0$ , $y_0$ and the properties of $q\left(x\right)$.

We consider the equation $$-r\left(x\right)y^{\text{'}}\left(x\right)+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ$$ where $f\in L_p\left(ℝ\right)$, $p\in [1,\infty ]$ ($L_{\infty }\left(ℝ\right):=C\left(ℝ\right)$) and $$0<r\in C^{}\left(ℝ\right),0\le q\in L_1^{}\left(ℝ\right).$$ We obtain minimal requirements to the functions $r$ and $q$, in addition to (), under which equation () is correctly solvable in $L_p\left(ℝ\right)$, $p\in [1,\infty ]$.

We consider the equation $$-y^{\prime \prime }(x)+q(x)y(x)=f(x),x\in ℝ,$$ where $f\in L_p(ℝ)$, $p\in [1,\mathrm{\infty }]$ ($L_{\mathrm{\infty }}(ℝ):=C(ℝ)$) and $$0\le q\in L_1^{\text{loc}}(ℝ);\exists a>0:\underset{x\in ℝ}{inf}{\int }_{x-a}^{x+a}q(t)dt>0,$$ (Condition (2) guarantees correct solvability of (1) in class $L_p(ℝ)$, $p\in [1,\mathrm{\infty }]$.) Let $y$ be a solution of (1) in class $L_p(ℝ)$, $p\in [1,\mathrm{\infty }]$, and $\theta$ some non-negative and continuous function in $ℝ$. We find minimal additional requirements to $\theta$ under which for a given $p\in [1,\mathrm{\infty }]$ there exists an absolute positive constant $c(p)$ such that the following inequality holds: $$\underset{x\in ℝ}{sup}\theta (x)|y(x)|\le c(p){\parallel f\parallel }_{L_p(ℝ)}\mathit{\hspace{1em}\hspace{1em}}\forall f\in L_p(ℝ).$$