\[
\boxed{
{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left( x \right) =-{\frac { \left( 2\,f \left( x \right)  \left( {\frac {\rm d}{{\rm d}x}}g \left( x \right)  \right) ^{2}g \left( x \right) - \left(  \left( g \left( x \right)  \right) ^{2}-1 \right)  \left( f \left( x \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}g \left( x \right) +2\, \left( {\frac {\rm d}{{\rm d}x}}f \left( x \right)  \right) {\frac {\rm d}{{\rm d}x}}g \left( x \right)  \right)  \right) {\frac {\rm d}{{\rm d}x}}y \left( x \right) }{f \left( x \right)  \left( {\frac {\rm d}{{\rm d}x}}g \left( x \right)  \right)  \left(  \left( g \left( x \right)  \right) ^{2}-1 \right) }}-{\frac { \left(  \left(  \left( g \left( x \right)  \right) ^{2}-1 \right)  \left(  \left( {\frac {\rm d}{{\rm d}x}}f \left( x \right)  \right)  \left( f \left( x \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}g \left( x \right) +2\, \left( {\frac {\rm d}{{\rm d}x}}f \left( x \right)  \right) {\frac {\rm d}{{\rm d}x}}g \left( x \right)  \right) -f \left( x \right)  \left( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f \left( x \right)  \right) {\frac {\rm d}{{\rm d}x}}g \left( x \right)  \right) - \left( 2\, \left( {\frac {\rm d}{{\rm d}x}}f \left( x \right)  \right) g \left( x \right) +v \left( v+1 \right) f \left( x \right) {\frac {\rm d}{{\rm d}x}}g \left( x \right)  \right) f \left( x \right)  \left( {\frac {\rm d}{{\rm d}x}}g \left( x \right)  \right) ^{2} \right) y \left( x \right) }{ \left( f \left( x \right)  \right) ^{2} \left( {\frac {\rm d}{{\rm d}x}}g \left( x \right)  \right)  \left(  \left( g \left( x \right)  \right) ^{2}-1 \right) }}=0}
\]