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Find a value of $v_0$ such that the solution of the initial value problem:

$$x''-x'-2x=0,\ x(0)=1,\ x'(0)=v_0$$

...is bounded at $t\rightarrow\infty$.

The solution $x(t)$ is bounded as $t\rightarrow\infty$ for all values of $v_0$.

No value of $v_0$ will make the solution $x(t)$ bounded as $t\rightarrow\infty$.

$v_0=-1$

$v_0=0$

$v_0=1$

$v_0=2$