= Solution
On an <interval> with $x>0$, use the <power-law ansatz> $y=x^r$. Substitution gives $r(r-1)+2r+1=r^2+r+1=0$, whose roots are $r=-1/2\pm i\sqrt3/2$. Taking real and imaginary parts of $x^r$ gives
$$
\boxed{y(x)=x^{-1/2}\left[C_1\cos\left(\frac{\sqrt3}{2}\log x\right)+C_2\sin\left(\frac{\sqrt3}{2}\log x\right)\right].}
$$
These two real solutions are <linearly independent>: their <Wronskian> is $(\sqrt3/2)x^{-2}\ne0$. They therefore span the solution space of the second-order <linear ordinary differential equation> on that <interval>.
For $x<0$, replace $x^{-1/2}$ by $|x|^{-1/2}$ and $\log x$ by $\log|x|$, with independent real constants on the negative <interval>. This follows equally by the change of variable $t=\log|x|$, which turns the <Cauchy-Euler equation> into $d^2y/dt^2+dy/dt+y=0$. The <coefficient> of $y''$ vanishes at zero, so the general formula is stated separately on either side. Every nonzero mode has an unbounded oscillatory envelope as $x\to0$; a continuous extension through zero requires the solution to vanish identically on both sides.
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