If

is it true that

I observed that if

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?

yert21trey123z7
2022-05-01
Answered

Second order differential inequality and comparison theorem.

If$x:[0,1]\Rightarrow \mathbb{R},x\in {C}^{\mathrm{\infty}}$ that satisfies

$$\{\begin{array}{l}\ddot{x}\le -2x\\ x(0)=x(1)=0\end{array}$$

is it true that$x\ge 0\text{}\text{on}\text{}[0,1]$ ?

I observed that if$x\ge 0$ , then $\ddot{x}\le 0$ , 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

$$\{\begin{array}{l}\ddot{x}=-2x\\ x(0)=x(1)=0\end{array}$$

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?

If

is it true that

I observed that if

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?

You can still ask an expert for help

Kendal Kelley

Answered 2022-05-02
Author has **16** answers

Answer: Concavity is enough:

$x\left(t\right)=x((1-t)\cdot 0+t\cdot 1)\ge (1-t)\cdot x\left(0\right)+t\cdot x\left(1\right)=0$ .

Alternatively, by Rolle's theorem,${x}^{\prime}\left({t}_{0}\right)=0$ for some ${t}_{0}\in (0,1)$ , ans as $x{}^{\u2033}\le 0\text{}\text{if}\text{}t{t}_{0}\text{}\text{and}\text{}{x}^{\prime}\left(t\right)\le 0\text{}\text{if}\text{}t{t}_{0}$ , so x not decreases on $[0,{t}_{0}]$ -thus is non-negative there, and x not increases on $[{t}_{0},1]$ - thus again is non-negative there.

Alternatively, by Rolle's theorem,

asked 2021-01-02

Find

asked 2022-03-25

How can be the state-space representation rewritten into differential equation?

$\dot{x}}_{1}=-{x}_{1}+{x}_{2$

${\dot{x}}_{2}=-{x}_{1}+u$

$A\left(\begin{array}{cc}-1& 1\\ -1& 0\end{array}\right)x{\textstyle \phantom{\rule{1em}{0ex}}}B\left(\begin{array}{c}0\\ 1\end{array}\right)u$

asked 2022-04-23

How to solve

$\ddot{\overrightarrow{u}}=\overrightarrow{u}\times \hat{k}$

asked 2022-03-23

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function, and suppose $f={f}^{\prime}$ and $f\left(0\right)=1$ . Then prove $f\left(x\right)\ne 0$ for all $x\in \mathbb{R}$

asked 2021-12-13

Find the differential.

a)$z=\frac{1}{2}({e}^{{x}^{2}+{y}^{2}}-{e}^{-{x}^{2}-{y}^{2}})$

b)$w=2{z}^{3}y\mathrm{sin}x$

a)

b)

asked 2022-04-13

Find the general solution

$x\frac{dy}{dx}-y=\frac{1}{{x}^{2}}$

asked 2022-03-16

Assessing stability or instability of a system of equations with complex eigenvalues

Having this system , we get clearly two complex eigenvalues. If one has to assess the stability of the system at these eigenvalues, we have for the matrix A:

$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$

that ${T}^{2}-4\mathrm{\Delta}\ge {\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}\le {\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}=0$, where $T=a+c$, while $\mathrm{\Delta}=ad-bc=DetA$. But with complex eigenvalues $\pm 2i$, T which must be real, becomes complex and $\delta$ is always 0, when it would vary from greater than or lesser than zero for real eigenvalues. How do we solve this with complex eigenvalues?