= Reduction of order across a zero of the known solution
The integral representation $y_2=y_1\int e^{-\int p}/y_1^2$ is derived where $y_1\ne0$. At a zero, its integral factor may have a pole while the product has a removable singularity. For regular coefficients, continue the actual second solution by its finite initial data and initial-value uniqueness; the <Abel identity> preserves a nonzero <Wronskian>. In the degree-one <Legendre differential equation>, $y_1=x$ and $y_2=x\operatorname{arctanh}x-1$ illustrate this cancellation at zero.
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