Differential equations MOC

Existence and uniqueness theorem for IVPs

In general, given an initial value problem

with initial conditions

the existence and uniqueness theorem guarantees the existence of a unique solution for initial conditions in the region for which the functions are real, finite, and continuous. This solution may only be defined for a small neighbourhood around the initial condition.

First order

Given an initial value problem

If and continuous¹ over some region where , then there must exist one and only one solution to the initial value problem.²³

This solution may be analytical or numerical, the only thing guaranteed is that some solution exists. Note that the EUT can not tell you when a solution does not exist, as the implication only goes one way. The EUT condition may not hold for an IVP that is solvable.

Practice problems

• 2017. Elementary differential equations and boundary value problems, p. 57 (§2.4 Problems)

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tidy | en | sembr | review

Footnotes

1. As well as real, finite, and single-valued. ↩

2. 2017. Elementary differential equations and boundary value problems, p. 51, p. 84 ↩

3. 2024. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering, §2.5, p. 29 ↩