= Delayed solutions of a square-root differential equation
{title2=$y'=2\sqrt{|y|},\quad y(0)=0$}
This <initial value problem> uses a <square root> and <absolute value> to define a <continuous function> on all of $\mathbb R^2$. For every finite waiting time $c\geq0$, a forward <solution of a differential equation> on $t\geq0$ is
$$
y_c(t)=\begin{cases}0,&0\leq t\leq c,\\(t-c)^2,&t>c.\end{cases}
$$
The constant <solution of a differential equation> $y_\infty(t)=0$ is another choice. For example, at $t=3$, waits $c=0,1,2,3$ give values $9,4,1,0$. These are <dimensionless> mathematical trajectories; no measured physical waiting time is being supplied.
Verification uses the original <ordinary differential equation>, including the join. Before departure, both $y_c'$ and $2\sqrt{|y_c|}$ vanish. For $t>c$, their common value is $2(t-c)$. For $c>0$, the difference quotient at the join is
$$
\frac{y_c(c+s)-y_c(c)}s=\begin{cases}0,&s<0,\\s,&s>0,\end{cases}\qquad\longrightarrow0.
$$
Thus the <derivative> exists at $c$, equals the rate law there, and is continuous across the join. For $c=0$ the same computation gives the right <derivative> at the starting endpoint. For $c>0$, the left and right <second derivatives> at departure are $0$ and $2$: these forward <solutions of a differential equation> are <continuously differentiable functions>, without being twice differentiable at departure. The local two-sided statement of the <Peano existence theorem> is separate from this explicitly forward family.
The rate law tends to zero as $y\to0$, but its secant slopes from zero are unbounded:
$$
\frac{|f(\varepsilon)-f(0)|}{|\varepsilon-0|}=\frac{2}{\sqrt\varepsilon}\longrightarrow\infty\qquad(\varepsilon\downarrow0).
$$
No finite local <Lipschitz constant> exists at zero. At $\varepsilon=10^{-2},10^{-4},10^{-6}$ the quotient is $20,200,2000$, even though the rates are shrinking to zero. <Continuity> controls the rate itself; <Lipschitz continuity> would control its change relative to the state displacement.
\Image[/peano-vector-field-regularity.png]
{title=A continuous square-root rate has unbounded secant slopes at zero}
{description=Left: the continuous rate $f(y)=2\sqrt{|y|}$ on both sides of zero, with two positive-state secants. Right: their slope $2/\sqrt\varepsilon$ on logarithmic axes, including the values $20,200,2000$.}
There is also a numerical lesson. The <Forward Euler method> starting exactly at zero gives $Y_{k+1}=Y_k+2\Delta t\sqrt{|Y_k|}=0$ at every step. It reproduces the constant <solution of a differential equation> for every positive <step size>, while the moving trajectories remain equally valid. A computation producing one trajectory cannot establish uniqueness. Likewise, the <separable differential equation> calculation that divides by $\sqrt{|y|}$ only describes the moving pieces: checking the original equation recovers the zero pieces and their joins.
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