# The significance of failure of uniqueness in differential equations

The nonlinear ODE: $y'(t)=y(t)^{1/2}$ with initial condition $y(0)=1$ has two solutions. Non-uniqueness is not surprising because of the failure of Lipschitz continuity in the $y$ term. While this is formally true, what if any, is the practical significance of the failure of uniqueness of solutions? For example, if this ODE (or a similar PDE) modelled some biological or physical phenomenon, would non-uniqueness mean anything?

Edit: Thanks for the responses so far! I found a passage in Fung and Tong's Classical and Computational Solid Mechanics in Chapter 21 on page 849 which reads:

One of the distinct characteristics of nonlinear problems is that the solution may not be unique. In the nonuniqueness, there lies much of nature’s secret. Examples in solid mechanics are the buckling of thin shells, self-equilibrating residual stresses and strains, three-dimensional solutions in bodies with apparently two-dimensional boundary conditions, and many problems in plasticity.

I am not so familiar with these models: perhaps someone wiser than myself might have some specific insight?

If $\gamma : I \rightarrow \mathbb{R}^3$ denotes the velocity of a particle moving through a fluid. Then in all likelihood $\gamma'(t) = F(\gamma(t))$ for some suitable function $F : \mathbb{R}^3 \rightarrow \mathbb{R}^3$. Now the velocity is uniformly Lipschitz continuous or the particle would be subject to a arbitrarily large acceleration/force. This is not something that occurs in real world. In other words, it is extremely likely that $F$ is Lipschitz continuous on its domain.