\[ -y'(x) \left (\frac {2 f'(x)}{f(x)}-\frac {g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}\right )+y(x) \left (-\frac {f''(x)}{f(x)}+\frac {f'(x) \left (\frac {2 f'(x)}{f(x)}-\frac {g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}\right )}{f(x)}-\frac {v^2 g'(x)^2}{g(x)^2}+g'(x)^2\right )+y''(x)=0 \] ✓ Mathematica : cpu = 0.162916 (sec), leaf count = 36
\[\left \{\left \{y(x)\to c_1 f(x) J_{\sqrt {v^2}}(g(x))+c_2 f(x) Y_{\sqrt {v^2}}(g(x))\right \}\right \}\] ✓ Maple : cpu = 0.105 (sec), leaf count = 20
\[\{y \left (x \right ) = \left (c_{1} \BesselJ \left (v , g \left (x \right )\right )+c_{2} \BesselY \left (v , g \left (x \right )\right )\right ) f \left (x \right )\}\]