Second order differential inequality and comparison theorem.If x:[0,1]→R,x∈C∞ that satisfies{x¨≤−2xx(0)=x(1)=0is it true that x≥0 on [0,1]?I observed that if x≥0, 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)=0then 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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Second order differential inequality and comparison theorem.If x:[0,1]→R,x∈C∞ that satisfies{x¨≤−2xx(0)=x(1)=0is it true that x≥0 on [0,1]?I observed that if x≥0, 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)=0then 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?

Avery Maxwell · Accepted Answer

Answer: Concavity is enough:x(t)=x((1−t)⋅0+t⋅1)≥(1−t)⋅x(0)+t⋅x(1)=0.Alternatively, by Rolle&#039;s theorem, x′(t0)=0 for some t0∈(0,1), ans as x″≤0 if t&amp;lt;t0 and x′(t)≤0 if t&amp;gt;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.

Second order differential inequality and comparison theorem. If

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