# Liouville's Theorem (Differential Equations)

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## Theorem

Let $\map \Phi t$ be a solution to the matrix differential equation:

- $X' = \map A t X$

with $\map A t$ continuous on the interval $I$ such that $t_0 \in I$.

Then:

- $\det \map \Phi t = e^{\int_{t_0}^t \tr \map A s \rd s} \det \map \Phi {t_0}$

## Proof

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## Source of Name

This entry was named for Joseph Liouville.