= Solution
Multiply the equation by $x$ to obtain $xy''-P(x)y'-Q(x)y=0$. The leading terms after a <Frobenius method> trial $y=x^r$ are
$$
\bigl(r(r-1)-P_0r\bigr)x^{r-1};
$$
the term involving $Q_0$ first appears at order $x^r$. The <indicial equation> is therefore
$$
r(r-1-P_0)=0,\qquad \boxed{r=0,\ P_0+1.}
$$
To obtain the <first correction to a regular singular solution by reduction of order>, start from the assumed solution $y_1=1+\beta x+O(x^2)$ and use the <Abel identity>:
$$
W=C\exp\left(\int\frac{P(x)}x\,dx\right)
=Cx^{P_0}\bigl(1+P_1x+O(x^2)\bigr).
$$
We work initially on $x>0$ near zero; since $P_0$ is a positive integer, the resulting expansion is an ordinary <power series>. Also $y_1^{-2}=1-2\beta x+O(x^2)$, so the <reduction of order> integral has integrand
$$
\frac{W}{y_1^2}=Cx^{P_0}\bigl(1+(P_1-2\beta)x+O(x^2)\bigr).
$$
Choose its integration constant to be zero, eliminating an added multiple of $y_1$. Integrating and multiplying by $y_1$ gives
$$
y_2=C\left[\frac{x^{P_0+1}}{P_0+1}
+\left(\frac{\beta}{P_0+1}+\frac{P_1-2\beta}{P_0+2}\right)x^{P_0+2}
+O(x^{P_0+3})\right].
$$
With $C=P_0+1$ the requested normalized answer is
$$
\boxed{y_2=x^{P_0+1}
+\frac{(P_0+1)P_1-P_0\beta}{P_0+2}x^{P_0+2}
+O(x^{P_0+3}).}
$$
Its nonzero <Wronskian> makes it independent of $y_1$. The existence of the asserted $y_1$ also forces $P_0\beta+Q_0=0$; there is no need to introduce $Q_0$ into the normalized second coefficient.
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