# 2.4-2.5 - Sections 2.4-2.5 Existence and uniqueness Theorem...

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Sections 2.4-2.5 Existence and uniqueness Theorem 2.4.1 is a simplifed version oF Theorem 2.4.2. Major di±erence being: existence and uniqueness is sort oF global For Theorem 2.4.1, but only guaranteed locally on a fxed interval ( t 0 h,t 0 + h )( h unspecifed) For Theorem 2.4.2. This is why nonlinear equations tend to be harder to analyze than linear equations in general. And linear equations will be our major concern For the remaining oF this semester. You can glance through page 69, it’s not that important. Pay attention to the remark Following Theorem 2.4.2, which states that existence is guaranteed locally on the basis oF the continuity oF f alone. Also pay attention to item 3 oF the summary on page 75. ²ollow these steps to approach an initial value problem dy dt = f ( t,y ) ,y ( t 0 )= y 0 . Step 1: Determine the interval oF continuity For f and ∂f ∂y . (Recall the defnition!) Step 2: Check where the given point ( t 0 ,y 0 ) lies. Dependent on the di±erent scenarios, you will draw di±erent conclusions.

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2.4-2.5 - Sections 2.4-2.5 Existence and uniqueness Theorem...

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