Solution
= Solution
For $x=e^z>0$, let $Y(z)=y(e^z)$. The <chain rule> gives
$$
y'(x)=\frac{Y'(z)}x,\qquad y''(x)=\frac{Y''(z)-Y'(z)}{x^2}.
$$
Multiplying each original <Cauchy-Euler equation> by $x^2$ therefore gives
$$
\boxed{\text{(1)}\quad Y''+3Y'+2Y=0,\qquad \alpha=3,\ \beta=2,}
$$
and
$$
\boxed{\text{(2)}\quad Y''+Y=0,\qquad \alpha=0,\ \beta=1.}
$$
The exponential change of variable has made both equations have constant coefficients.