Vault Of Euclid

Uniqueness of Solutions to f'(x) = f(x)

Logarithmic and exponential functions core

Prove that the only solutions of the differential equation \(f'(x) = f(x)\) are the functions \(Ce^x\), where \(C\) is a constant.

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What you need first

the product rule for differentiation
the derivative of the exponential function
that a function with a zero derivative is constant

What this problem tests

Tests the ability to use substitution and the product rule to solve a simple differential equation and prove uniqueness.

This is a fundamental result in calculus and differential equations, establishing the uniqueness of the exponential growth and decay models.

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