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Miklós Schweitzer
2011 Miklós Schweitzer
9
differential equation
differential equation
Source: miklos schweitzer 2011 q9
August 29, 2021
differential equation
limit
real analysis
Problem Statement
Let
x
:
[
0
,
∞
)
→
R
x: [0, \infty) \to\Bbb R
x
:
[
0
,
∞
)
→
R
be a differentiable function. Prove that if for all t>1
x
′
(
t
)
=
−
x
3
(
t
)
+
t
−
1
t
x
3
(
t
−
1
)
x'(t)=-x^3(t)+\frac{t-1}{t}x^3(t-1)
x
′
(
t
)
=
−
x
3
(
t
)
+
t
t
−
1
x
3
(
t
−
1
)
then
lim
t
→
∞
x
(
t
)
=
0
\lim_{t\to\infty} x(t) = 0
lim
t
→
∞
x
(
t
)
=
0
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