Delayed solutions of a square-root differential equation ($y^{\prime }=2\sqrt{|y|},y(0)=0$)

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 $c\ge 0$, a forward solution of a differential equation on $t\ge 0$ isThe 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 $y_c^{\prime }$ 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 isThus 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 $y\to 0$, but its secant slopes from zero are unbounded:No finite local Lipschitz constant exists at zero. At $\epsilon =10^{-2},10^{-4},10^{-6}$ 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.

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.

For the autonomous differential equation $v^{\prime }=\sqrt{1-v^2}$, the real solutions are nondecreasing and lie in $[-1,1]$. On each nonconstant interval, separation of variables gives $\arcsin v=t-C$, so the solution crosses from $-1$ to $1$ in exactly time $\pi$. Every nonconstant solution on the entire real time axis is consequentlyfor some real $a$. The two constant solutions $v\equiv \pm 1$ complete the list; solutions on shorter intervals are restrictions of these. At the joins both the derivative and the right-hand side are zero, so these are continuously differentiable functions. The right-hand side is continuous but fails Lipschitz continuity at both endpoints, allowing arbitrary waiting time at $-1$ before departure. Once $v$ reaches $1$, monotonicity forces it to stay there. The sine representation without the nonnegative-cosine restriction would incorrectly introduce decreasing pieces.