In the study of ordinary differential equations (ODEs) of the form $hat{L}[y(x)]=f(x)$, where $hat{L}$ is a linear differential operator, two related phenomena can arise: resonance, where $f(x)propto u(x)$ and $hat{L}[u(x)]=0$, and repeated roots, wh
ere $f(x)=0$ and $hat{L}=hat{D}^n$ for $ngeq 2$. We illustrate a method to generate exact solutions to these problems by taking a known homogeneous solution $u(x)$, introducing a parameter $epsilon$ such that $u(x)rightarrow u(x;epsilon)$, and Taylor expanding $u(x;epsilon)$ about $epsilon = 0$. The coefficients of this expansion $frac{partial^k u}{partialepsilon^k}big{|}_{epsilon=0}$ yield the desired resonant or repeated root solutions to the ODE. This approach, whenever it can be applied, is more insightful and less tedious than standard methods such as reduction of order or variation of parameters. While the ideas can be introduced at the undergraduate level, we could not find any elementary or advanced text that illustrates these ideas with appropriate generality.
Liquid-liquid phase separation occurs not only in bulk liquid, but also on surfaces. In physiology, the nature and function of condensates on cellular structures remain unexplored. Here, we study how the condensed protein TPX2 behaves on microtubules
to initiate branching microtubule nucleation, which is critical for spindle assembly in eukaryotic cells. Using fluorescence, electron, and atomic force microscopies and hydrodynamic theory, we show that TPX2 on a microtubule reorganizes according to the Rayleigh-Plateau instability, like dew droplets patterning a spider web. After uniformly coating microtubules, TPX2 forms regularly spaced droplets from which branches nucleate. Droplet spacing increases with greater TPX2 concentration. A stochastic model shows that droplets make branching nucleation more efficient by confining the space along the microtubule where multiple necessary factors colocalize to nucleate a branch.
