Differential equations MOC

Existence and uniqueness theorem for IVPs

In general, given an initial value problem

𝑑𝑛𝑦𝑑𝑥𝑛=𝑓(𝑥,𝑦(0),…𝑦(𝑛−1))

with initial conditions

𝑦(𝑥0)=𝑦0𝑦(1)(𝑥0)=𝑦(1)0⋮𝑦(𝑛−1)(𝑥0)=𝑦(𝑛−1)0

the existence and uniqueness theorem guarantees the existence of a unique solution for initial conditions in the region for which the functions 𝑓,𝜕𝑓𝜕𝑦,…,𝜕𝑓𝜕𝑦(𝑛−1) 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

𝑑𝑦𝑑𝑥=𝑓(𝑥,𝑦)∧𝑦(𝑥0)=𝑦0

If 𝑓(𝑥,𝑦) and 𝜕𝑓𝜕𝑦(𝑥,𝑦) continuous¹ over some region 𝑅 ⊆ℝ2 where (𝑥0,𝑦0) ∈𝑅, 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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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 ↩