= First correction to a regular singular solution by reduction of order
{title2=$y_2=x^{P_0+1}+\frac{(P_0+1)P_1-P_0\beta}{P_0+2}x^{P_0+2}+\cdots$}
Suppose $y''-P(x)y'/x-Q(x)y/x=0$ has analytic $P,Q$ near zero, $P(x)=P_0+P_1x+\cdots$, and a solution $y_1=1+\beta x+\cdots$, with $P_0$ a positive integer. The <indicial roots> are $0,P_0+1$. The <Abel identity> gives $W=Cx^{P_0}(1+P_1x+\cdots)$. Dividing by $y_1^2$ gives $Cx^{P_0}[1+(P_1-2\beta)x+\cdots]$. Integrating this <reduction of order> expression from zero, multiplying by $y_1$, and choosing $C=P_0+1$ proves the displayed normalized expansion. The integral's zero constant eliminates the added multiple of $y_1$. Its nonzero <Wronskian> proves independence. In this setting the assumed exponent-zero solution prevents a logarithm in the constructed exponent-$P_0+1$ solution.
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