Second order differential inequality and comparison theorem. If x:[0,1] \rightarrow \mathbb R

yert21trey123z7 2022-05-01 Answered
Second order differential inequality and comparison theorem.
If x:[0,1]R,xC that satisfies
{x¨2xx(0)=x(1)=0
is it true that x0 on [0,1]?
I observed that if x0, then x¨0, so x is concave. However I could not find out any other properties. I think if there exists a solution of the following differential equation
{x¨=2xx(0)=x(1)=0
then some kind of comparative theorem can be used. However, there is no non-trivial solution for this.
If the above proposition is incorrect, could you give me a counterexample?
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Answers (1)

Kendal Kelley
Answered 2022-05-02 Author has 16 answers
Answer: Concavity is enough:
x(t)=x((1t)0+t1)(1t)x(0)+tx(1)=0.
Alternatively, by Rolle's theorem, x(t0)=0 for some t0(0,1), ans as x0 if t<t0 and x(t)0 if t>t0, so x not decreases on [0,t0]-thus is non-negative there, and x not increases on [t0,1] - thus again is non-negative there.
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