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The exponential is the unique solution of $y^{'} = y$ with $y (0) = 1$

Statement

If $y : R \to R$ is differentiable, $y^{'} = y$ , and $y (0) = 1$ , then $y = \exp$ .

Facts & Assumptions

Given: A differentiable solution $y$ of the initial-value problem.

[L1]

$\exp^{'} = \exp$ , $\exp (- x) = 1 / \exp (x)$ , and the series definition gives $\exp (0) = 1$ (The exponential function is smooth and $(\exp)^{'} = \exp$ , The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , The real exponential function and the number $e$ by a power series).

[L2]

Products and composites satisfy Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ and The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ .

[L3]

Differentiability implies continuity, and a continuous function with zero derivative on an interval is constant (A function differentiable at $c$ is continuous at $c$ , A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

Proof

technique · direct

1.1

Define $h (x) = y (x) \exp (- x)$ . By [L1] and [L2], $h^{'} (x) = y^{'} (x) \exp (- x) - y (x) \exp (- x) = 0$ .

givenL1L2

2.1

The differentiable function $h$ is continuous, so [L3] makes it constant; $h (0) = y (0) \exp (0) = 1$ .

step 1.1L1L3

3.1

Thus $y (x) \exp (- x) = 1$ , and multiplying by $\exp (x)$ gives $y (x) = \exp (x)$ .

step 2.1L1algebra∎

Depends on

Used by

Dependency tree · two levels

31 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
J. K. Hunter, An Introduction to Real Analysis, Chapter 10 (standard reference, not scraped)
University of Pennsylvania MATH 3600, Section 34 (standard reference, not scraped)

The exponential is the unique solution of y′=yy′=y with y(0)=1y(0)=1