\[ -y'(x) \left (\frac {(2 v-1) g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}+\frac {2 h'(x)}{h(x)}\right )+y(x) \left (g'(x)^2+\frac {h'(x) \left (\frac {(2 v-1) g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}+\frac {2 h'(x)}{h(x)}\right )}{h(x)}-\frac {h''(x)}{h(x)}\right )+y''(x)=0 \] ✗ Mathematica : cpu = 0.897719 (sec), leaf count = 0 , could not solve
DSolve[-(Derivative[1][y][x]*(((-1 + 2*v)*Derivative[1][g][x])/g[x] + (2*Derivative[1][h][x])/h[x] + Derivative[2][g][x]/Derivative[1][g][x])) + y[x]*(Derivative[1][g][x]^2 + (Derivative[1][h][x]*(((-1 + 2*v)*Derivative[1][g][x])/g[x] + (2*Derivative[1][h][x])/h[x] + Derivative[2][g][x]/Derivative[1][g][x]))/h[x] - Derivative[2][h][x]/h[x]) + Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 0.101 (sec), leaf count = 24
\[ \left \{ y \left ( x \right ) =h \left ( x \right ) \left ( g \left ( x \right ) \right ) ^{v} \left ( {{\sl Y}_{v}\left (g \left ( x \right ) \right )}{\it \_C2}+{{\sl J}_{v}\left (g \left ( x \right ) \right )}{\it \_C1} \right ) \right \} \]