Let $\lambda$ be the first positive value for which $y=0$ where $y(x)$ satisfy the following differential equation $ y''+\frac{2}{x}y'+y^n=0,\qquad\text{where }n\in\mathbb{R},\ y(0)=1,\text{ and }\ y'(0)=0. $ This equation is known as the Lane-Emden equation and has analytic solutions for $n=0,1,5$.
For $n=0$, $ y(x)=1-\frac{x^2}{6}\Rightarrow\lambda=\sqrt{6} $
For $n=1$, $ y(x)=\frac{\sin{x}}{x}\Rightarrow\lambda=\pi $
And for $n = 5$, $ y(x)=\frac{1}{\sqrt{1+\frac{x^2}{3}}}\Rightarrow\lambda=\infty $
We want to show that $\lambda$ increases steadily as $n$ goes from $0$ to $5$. We can verify numerically that it is indeed true but it has not yet been proved analytically.