\[ -y'(x) \left (\frac {2 f'(x)}{f(x)}-\frac {g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}\right )+y(x) \left (-\frac {f''(x)}{f(x)}+\frac {f'(x) \left (\frac {2 f'(x)}{f(x)}-\frac {g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}\right )}{f(x)}-\frac {v^2 g'(x)^2}{g(x)^2}+g'(x)^2\right )+y''(x)=0 \] ✗ Mathematica : cpu = 0.933138 (sec), leaf count = 0 , could not solve
DSolve[-(Derivative[1][y][x]*((2*Derivative[1][f][x])/f[x] - Derivative[1][g][x]/g[x] + Derivative[2][g][x]/Derivative[1][g][x])) + y[x]*(Derivative[1][g][x]^2 - (v^2*Derivative[1][g][x]^2)/g[x]^2 - Derivative[2][f][x]/f[x] + (Derivative[1][f][x]*((2*Derivative[1][f][x])/f[x] - Derivative[1][g][x]/g[x] + Derivative[2][g][x]/Derivative[1][g][x]))/f[x]) + Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 0.109 (sec), leaf count = 20
\[ \left \{ y \left ( x \right ) =f \left ( x \right ) \left ( {{\sl Y}_{v}\left (g \left ( x \right ) \right )}{\it \_C2}+{{\sl J}_{v}\left (g \left ( x \right ) \right )}{\it \_C1} \right ) \right \} \]