= Limitations of integrating factors for second-order equations
{title2=$w=\exp\!\int(B-A')/A\,dx$}
On an interval where $A\ne0$, the displayed factor gives
$$
w(Ay''+By'+Cy)=(wAy')'+wCy.
$$
This follows from $(wA)'=wB$. It gives a useful self-adjoint form but does not solve a general second-order <ordinary differential equation>, since the $wCy$ term remains. If $C=0$, it does reduce to one <integration> for $y'$. Another reducible class has $C=B'-A''$, because the original left side is then $[Ay'+(B-A')y]'$. These are sufficient special cases, not a classification of all solvable equations.
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