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Differential Equations

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Difficult

Set of All Solutions that can be Generated

DIFFEQ-WIN0X5

Consider the following initial value problem:

$$\begin{eqnarray*}u_1'&=&u_2,\ u_1(0)=x\\\ \\\ u_2'&=&u_1,\ u_2(0)=y\end{eqnarray*}$$

The initial values $x$ and $y$ are to be chosen at random.

Consider the set of all solutions that can be generated by choosing $x$ and $y$.

Choose the BEST answer.

A

Every solution $u=u(t)$ satisfies $\lim_{t\rightarrow\infty}\|u(t)\|=0$.

B

Almost every solution $u=u(t)$ satisfies
$\lim_{t\rightarrow\infty}\|u(t)\|=0$. In other words, when $x$ and $y$ are
chosen at random, there is a high probability that
$\lim_{t\rightarrow\infty}\|u(t)\|=0$ holds.

C

About half of the solutions satisfy $\lim_{t\rightarrow\infty}\|u(t)\|=0$
and about half satisfy $\lim_{t\rightarrow\infty}\|u(t)\|=\infty$.

D

Almost every solution $u=u(t)$ satisfies
$\lim_{t\rightarrow\infty}\|u(t)\|=\infty$. In other words, when $x$ and $y$
are chosen at random, there is a high probability that
$\lim_{t\rightarrow\infty}\|u(t)\|=\infty$ holds.

E

Every solution $u=u(t)$ satisfies
$\lim_{t\rightarrow\infty}\|u(t)\|=\infty$.