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Differential Equations

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Moderate

Solutions that Approach Zero at Infinity

DIFFEQ-E9N1EZ

Consider the equation $y''+(2r-1)y'+r(r-1)y=0$ where $r\in\mathbb{R}$ is a real number.

Find the range of $r$ such that all solutions approach zero as $x\to+\infty$.

A

Empty set $\emptyset$, since $r=1/2,0$ and $y=C_1e^{x/2}+C_2$ doesn't approach zero necessarily.

B

$(-\infty, 0)$

C

$(-\infty, 1)$

D

$(-\infty, 1]$

E

$(-\infty, 0]$