Two theorems, two different promises
By now this rung has handed you two results that sound similar but promise very different things. The Peano existence theorem says: if the right-hand side f(t, y) of y' = f(t, y) is merely continuous near the starting point, then at least one solution exists. The Picard-Lindelof theorem asks for more — a Lipschitz condition on f — and in return gives you more: exactly one solution. The gap between those two promises is the whole subject of this guide. Existence is almost free; uniqueness must be earned.
It is tempting to assume the two always travel together — that wherever a solution exists, it must be the only one. That assumption is false, and learning exactly why is one of the genuine turning points in understanding differential equations. A near-beginner often pictures an initial value problem as a maze with one exit: pick a starting point, and the equation marches you forward along a single forced path. That picture is right when f is Lipschitz. Strip the Lipschitz condition away and the maze can branch.
The famous counterexample: y' = y^(2/3)
Consider the initial value problem y' = y^(2/3) with y(0) = 0. The right-hand side f(y) = y^(2/3) is perfectly continuous everywhere — there is no division by zero, no jump, nothing alarming — so Peano cheerfully guarantees a solution exists. And one does: the flat line y(t) = 0 satisfies it, since both sides are zero for all t. That is a constant solution, the kind separation of variables loves to overlook.
But the flat line is not alone. Separate the variables — integral of y^(-2/3) dy = integral of dt — and you get 3 y^(1/3) = t + C, so a second solution is y(t) = (t/3)^3 = t^3/27. Check it: this curve also starts at y(0) = 0, yet it climbs away from zero immediately. So the SAME starting point (0, 0) admits two completely different futures: stay flat forever, or lift off as a cubic. Two solutions through one point is already a violation of uniqueness — and we are only getting started.
Where exactly the proof breaks
Recall from the Lipschitz guide what the condition actually buys you. A Lipschitz bound says |f(t, y1) - f(t, y2)| is at most L |y1 - y2| — the slope of f in the y-direction can never exceed some fixed number L. That single bound is exactly the ingredient that makes the Picard iteration a contraction, so its successive approximations close in on one and only one fixed point. No Lipschitz bound, no contraction; no contraction, no guarantee that the iterates converge to a single limit.
For f(y) = y^(2/3) near y = 0, no such L exists. Take y2 = 0 and let y1 shrink toward zero: the ratio |y1^(2/3) - 0| / |y1 - 0| = y1^(2/3) / y1 = y1^(-1/3) grows without bound as y1 goes to 0. No finite L can sit above all those ratios, so f is not Lipschitz at the origin — and that is the single hinge on which everything turns. Picard-Lindelof simply does not apply here; it never promised anything, so nothing it said is broken. What broke is the unspoken expectation that uniqueness comes for free.
Not two solutions — infinitely many
The branching is far worse than a coin flip between flat and cubic. Here is the trick that detonates it. Pick any waiting time a greater than or equal to 0. Build a solution that lies flat on the t-axis until time a, then peels off as a shifted cube: y(t) = 0 for t at most a, and y(t) = ((t - a)/3)^3 for t beyond a. Each such curve is a genuine solution — it sits on the equilibrium for a while, and because the slope and the value both vanish smoothly at the lift-off point, the two pieces join without any kink.
Three of the infinitely many solutions through (0, 0):
a = 0 y(t) = (t/3)^3 lifts off immediately
a = 1 y(t) = 0, t <= 1 waits, then lifts off at t = 1
y(t) = ((t-1)/3)^3, t > 1
a = inf y(t) = 0 for all t never lifts off (the flat line)
Every choice a >= 0 gives a different valid solution.Since a can be any nonnegative number, there are infinitely many solutions through the single point (0, 0) — a continuum of them, one for each choice of when to leave the axis. This is the picture worth carrying away: failure of uniqueness is not a hair-thin technicality where two curves barely disagree. When it fails, it can fail enormously, with a flood of solutions pouring out of one initial condition. The flat line, by the way, is exactly the singular solution that envelopes the whole family — a solution that no single value of the integration constant in y = (t/3 + C/3)^3 can reach.
Why this is not a contrived edge case
It would be comforting to file y' = y^(2/3) under 'pathological curiosities' and move on. Resist that. The same square-root and cube-root growth shows up in real models: a leaking tank whose outflow follows Torricelli's law, y' proportional to the square root of y, fails to be Lipschitz right at the empty state y = 0 — which is exactly why 'when did the tank finish draining' can have no unique answer running backward in time. Fractional powers, absolute values, and other non-smooth right-hand sides are common, and every one of them is a place to check the slope before trusting uniqueness.
There is also a deeper lesson about modelling honesty. If you are building a model of something physical and your equation loses uniqueness, the mathematics is telling you the model is under-determined: the present state genuinely does not fix the future, and you need extra physics to choose a branch. That is not a failure of the math — it is the math doing its job, flagging a place where your assumptions ran out. Uniqueness is tied to determinism, and where it breaks, determinism is making a real claim about the world that may simply not hold.
So hold the two promises clearly apart. Peano gives you a solution from continuity alone; Picard-Lindelof upgrades that to THE solution, but only by paying the Lipschitz price. Whenever you see a non-smooth right-hand side, slow down and ask whether the slope stays bounded. The next and final guide of this rung takes the surviving solution and asks a different question — not whether it is unique, but how far in time it actually reaches before it can blow up or run out.