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Nonlinear First-order Equations II
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Existence and Uniqueness
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True-or-False
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Existence and Uniqueness
True-or-False
1 .
An equation has a
unique local solution
if there is some interval I containing t
0
on which there is exactly one solution y.
[
Hint
]
True
False
2 .
If there is a unique solution on I, then there
is not
a unique solution on any subinterval I.
[
Hint
]
True
False
3 .
A
maximal solution
is a soultion whose only continuation is itself.
[
Hint
]
True
False
4 .
The concept of uniqueness is closely tied to the local nature of the solution.
[
Hint
]
True
False
5 .
Any
continuation
of y defined on an interval I, must equal y on this interval.
[
Hint
]
True
False
6 .
The graph of any continuation of y is a subset of the graph of y.
[
Hint
]
True
False
7 .
A solution defined on the whole real line is called a
local solution
.
[
Hint
]
True
False
8 .
The solution
normally means the unique maximal solution.
[
Hint
]
True
False
9 .
Theorem 1 is our
local existence and uniqueness theorem
.
[
Hint
]
True
False
10 .
Theorem 1 requires the function and the derivative to be continuous on some open rectangle.
[
Hint
]
True
False
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