Existence, Uniqueness, and Euler's Method

Most first-order initial value problems have no elementary solution, so two questions come before any attempt to solve one: whether a solution exists, and whether it is unique. Both matter: a model with no solution predicts nothing, and a model with several solutions predicts nothing definite. The existence-uniqueness theorem answers both under mild hypotheses, and its proof supplies a method — Picard iteration — that also underlies the numerical approach when no formula is available.

The existence and uniqueness theorem

For the general nonlinear problem the guarantee is local and rests on the continuity of $f$ together with its partial derivative in $y$ .

Two features distinguish this from the linear theorem. First, it is local: the solution is promised only on some interval $∣ t - t_{0} ∣ < h$ , possibly small, and the theorem gives no estimate of $h$ . Second, the hypotheses are sufficient but not necessary. Continuity of $f$ alone secures existence; uniqueness needs the extra control on $\partial f / \partial y$ . When the equation is linear, $f = - p (t) y + g (t)$ has $\partial f / \partial y = - p (t)$ , and the two continuity requirements reduce to continuity of $p$ and $g$ , recovering the stronger global linear statement.

A geometric corollary of uniqueness: two solution curves cannot intersect. A crossing point would be an initial condition satisfied by two different solutions, contradicting the theorem. This is what forbids solutions from touching an equilibrium in finite time in the phase-line analysis.

When uniqueness fails

The condition on $\partial f / \partial y$ cannot be dropped. Consider

y^{'} = y^{1/3}, y (0) = 0.

Here $f = y^{1/3}$ is continuous everywhere, so a solution exists, but $\partial f / \partial y = \frac{1}{3} y^{- 2/3}$ blows up at $y = 0$ — exactly the initial point. Uniqueness is not guaranteed, and indeed it fails. Separating variables gives $y = (\frac{2}{3} t)^{3/2}$ for $t \geq 0$ , but the constant function $y \equiv 0$ is also a solution, as is its negative. Worse, for any $t_{0} > 0$ the function that stays at zero until $t_{0}$ and then lifts off along $\pm (\frac{2}{3} (t - t_{0}))^{3/2}$ is a solution too. Infinitely many solution curves pass through the origin, fanning out into a solution funnel.

When the hypothesis on the y-derivative fails, uniqueness can fail: infinitely many solutions of y' = y^{1/3} leave the origin, some immediately, others after resting at zero for a while.

Away from the $t$ -axis the derivative $\partial f / \partial y$ is finite and continuous, so through any point not on the axis there is a unique solution. The failure is confined to where the hypothesis breaks.

The interval of existence

The number $h$ in the theorem is genuinely local, and its size can be estimated from a bound on $f$ . Suppose $f$ and $\partial f / \partial y$ are continuous on the rectangle $R : ∣ t ∣ \leq a$ , $∣ y ∣ \leq b$ (translating the initial point to the origin), and let $M$ bound $∣ f ∣$ on $R$ . Each iterate in the construction below has slope at most $M$ in magnitude, so its graph stays inside a bow-tie region of slopes $\pm M$ through the origin. That region remains inside $R$ as long as $∣ t ∣ \leq b / M$ , giving

h = min (a, \frac{b}{M}) .

The solution is certain to exist at least on $∣ t ∣ < h$ . It may extend much further, but the theorem alone promises only this.

The existence rectangle R and the bow-tie of slope M through the origin; the solution is guaranteed as far as the bow-tie stays inside R, which fixes h = min(a, b/M).

Picard's successive approximations

The proof of existence is constructive. Integrating $y^{'} = f (t, y)$ from the initial point converts the initial value problem $y^{'} = f (t, y)$ , $y (0) = 0$ into the equivalent integral equation

ϕ (t) = 0 \int t f (s, ϕ (s)) d s .

Any solution of one solves the other. Picard's idea is to solve the integral equation by iteration: start from a guess $ϕ_{0} (t) = 0$ and feed each approximation back through the integral to produce the next.¹

ϕ_{n + 1} (t) = 0 \int t f (s, ϕ_{n} (s)) d s, n = 0, 1, 2, \dots

Each iterate satisfies the initial condition; none generally satisfies the equation, but the sequence converges to one that does.

Picard iterates for y' = 2t(1+y): each successive approximation tracks the true solution over a wider interval, converging to it everywhere.

The convergence proof rests on a Lipschitz condition: if $\partial f / \partial y$ is bounded by $K$ on the rectangle, then $∣ f (t, y_{1}) - f (t, y_{2}) ∣ \leq K ∣ y_{1} - y_{2} ∣$ , which controls how fast successive iterates separate and forces the series of differences to converge. This same condition can replace continuity of $\partial f / \partial y$ in the theorem's hypotheses.

Euler's method

Picard iteration proves solutions exist but rarely computes them past a few steps. For actual numbers, Euler's method (1768) turns the tangent line into a stepping rule. The solution through $(t_{0}, y_{0})$ has known slope $f (t_{0}, y_{0})$ there, so its tangent line

y = y_{0} + f (t_{0}, y_{0}) (t - t_{0})

approximates it for a short step. Advancing to $t_{1} = t_{0} + h$ gives an approximate value $y_{1}$ ; from $(t_{1}, y_{1})$ the slope $f (t_{1}, y_{1})$ gives the next segment, and so on. With uniform step $h$ the rule is

y_{n + 1} = y_{n} + f (t_{n}, y_{n}) h, n = 0, 1, 2, \dots

The result is a polygonal path of linked tangent segments approximating the true curve.

Euler's method links tangent segments: at each step the slope f(t_n, y_n) extends the path by one interval, and the polygon drifts from the true solution as errors accumulate.

Worked example. Apply Euler's method to $y^{'} = 3 - 2 t - 0.5 y$ , $y (0) = 1$ , with step $h = 0.2$ . Each step reads the slope $f (t_{n}, y_{n}) = 3 - 2 t_{n} - 0.5 y_{n}$ and advances $y_{n + 1} = y_{n} + h f (t_{n}, y_{n})$ . The first slope is $f (0, 1) = 2.5$ , so $y_{1} = 1 + 0.2 (2.5) = 1.5$ ; then $f (0.2, 1.5) = 1.85$ gives $y_{2} = 1.5 + 0.2 (1.85) = 1.87$ , and so on. The exact solution $y = 14 - 4 t - 13 e^{- t /2}$ supplies the error at each node.

$t_{n}$	slope $f (t_{n}, y_{n})$	Euler $y_{n}$	exact $y (t_{n})$	error
$0.0$	$2.500$	$1.0000$	$1.0000$	$0.0000$
$0.2$	$1.850$	$1.5000$	$1.4371$	$0.0629$
$0.4$	$1.265$	$1.8700$	$1.7565$	$0.1135$
$0.6$	$0.739$	$2.1230$	$1.9694$	$0.1536$
$0.8$	$0.265$	$2.2707$	$2.0858$	$0.1849$
$1.0$	$-$	$2.3236$	$2.1151$	$0.2085$

The error grows at every node: the global truncation error accumulates in proportion to the number of steps taken.

Two kinds of error accumulate:

Local truncation error is the mistake made in a single step, from replacing the curve by its tangent. For Euler's method it is proportional to $h^{2}$ .
Global truncation error is the total deviation at a fixed $t$ after many steps. Because the number of steps grows as $1/ h$ , the global error is proportional to $h$ : Euler's method is first order. Halving the step size roughly halves the error.

First-order accuracy is coarse. The Runge-Kutta methods sample the slope at several points per step to reach far higher order at comparable cost, but every one of them is a refinement of the tangent-stepping idea Euler introduced.

Summary

The existence-uniqueness theorem guarantees a single local solution when $f$ and $\partial f / \partial y$ are continuous near the initial point; continuity of $f$ alone gives existence but not uniqueness.
Uniqueness can genuinely fail: $y^{'} = y^{1/3}$ , $y (0) = 0$ has infinitely many solutions, precisely because $\partial f / \partial y$ is unbounded there.
The interval of existence is only local, bounded below by $h = min (a, b / M)$ from a bound $M$ on $f$ over a rectangle.
Picard iteration proves existence by converging the sequence $ϕ_{n + 1} = \int_{0}^{t} f (s, ϕ_{n}) d s$ to the solution, under a Lipschitz condition.
Euler's method computes solutions numerically by linked tangent steps $y_{n + 1} = y_{n} + f (t_{n}, y_{n}) h$ , with global error proportional to $h$ .

Simmons, Differential Equations, Ch. 13 §69–§70 — the method of successive approximations and Picard's theorem: reformulating the initial value problem as an integral equation and iterating to the solution under a Lipschitz condition. The existence rectangle, non-uniqueness example, and Euler's method follow Boyce §2.8, §2.4, and §2.7. ↩