Sections 2.42.5
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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 Spring '08
 Fonken
 Derivative, Elementary algebra, Convex function, Concave function

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