AI 中文总结
本文针对源于天体物理学的Emden-Fowler方程，结合分析与数值方法研究其解的参数依赖关系，重点估计解的第一个零点，并用所得结果反向求解方程及判定超定边值问题的可解条件。
AI 中文摘要
我们研究源于天体物理学的Emden-Fowler方程$u''=-x^ru^p$，其初值条件为$u(0)=1$、$u'(0)=0$，分析其解对参数的定性与定量依赖关系，并结合数值模拟开展研究。重点关注根据参数$r$、$p$估计解$u$的第一个零点$x_0$。所得结果用于解决两个新问题：一是从$x_0$出发反向求解Emden-Fowler方程；二是将其转化为超定边值问题，并判定该问题可解的条件。
英文摘要
We investigate the Emden-Fowler equation $u''=-x^ru^p, u(0)=1, u'(0)=0$, with roots in astrophysics, and study the qualitative and quantitative dependence of its solution on the parameters; the analytical work is paralleled by numerical simulations. A special attention is given to estimating the first zero $x_0$ of $u$ in terms of $r, p$. The results are used to address two apparently new issues: first, we solve EF backwards starting from $x_0$; second, we transform it into an overdetermined boundary value problem and decide when is this solvable.
