= Solution
The homogeneous equation is a <Cauchy-Euler equation>. With the <power-law ansatz> $y=x^r$ on $x>0$, its <indicial equation> is
$$
r(r-1)+\alpha r+\frac{(\alpha-1)^2}{4}=\left(r+\frac{\alpha-1}{2}\right)^2=0.
$$
Thus $r=(1-\alpha)/2$ is a repeated <indicial root>. The two independent solutions are $x^r$ and $x^r\log x$, giving
$$
\boxed{y_h(x)=x^{(1-\alpha)/2}(C_1+C_2\log x),\qquad x>0.}
$$
Their <Wronskian>, with $W=y_1y_2'-y_1'y_2$, is $x^{2r-1}=x^{-\alpha}\ne0$, verifying <linear independence>.
Back to article page
