Limitations of integrating factors for second-order equations ($w=\exp \int (B-A^{\prime })/Adx$)

On an interval where $A\ne 0$, the displayed factor givesThis follows from $(wA{)}^{\prime }=wB$. 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 $y^{\prime }$. Another reducible class has $C=B^{\prime }-A^{\prime \prime }$, because the original left side is then $[A{y}^{\prime }+(B-A^{\prime })y{]}^{\prime }$. These are sufficient special cases, not a classification of all solvable equations.