= Solution
Define the <Wronskian> by $W=y_1y_2'-y_1'y_2$. Differentiating and using the normal-form <differential equation> gives
$$
W'=y_1y_2''-y_1''y_2=-P(x)(y_1y_2'-y_1'y_2)=-P(x)W.
$$
Integrating this first-order <linear differential equation> proves <Abel's identity>:
$$
\boxed{W(x)=C\exp\left[-\int^x p(u)\,du\right],\qquad p(u)=\frac1u+\frac1{u-m}.}
$$
Thus, on any interval avoiding zero and $m$,
$$
\boxed{W(x)=\frac{C}{x(x-m)}}.
$$
Absolute-value signs from real logarithms can be absorbed into the constant on each such interval. For <linearly independent> solutions, $C\ne0$.
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