\[ y(x) \left (\frac {f'(x) \left (\frac {2 f'(x)}{f(x)}+\frac {g''(x)}{g'(x)}-\frac {g'(x)}{g(x)}\right )}{f(x)}-\frac {f''(x)}{f(x)}-\frac {v^2 g'(x)^2}{g(x)^2}+g'(x)^2\right )-y'(x) \left (\frac {2 f'(x)}{f(x)}+\frac {g''(x)}{g'(x)}-\frac {g'(x)}{g(x)}\right )+y''(x)=0 \] ✓ Mathematica : cpu = 0.152243 (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 = 0.178 (sec), leaf count = 20
\[ \left \{ y \left ( x \right ) =f \left ( x \right ) \left ( {{\sl Y}_{v}\left (g \left ( x \right ) \right )}{\it \_C2}+{{\sl J}_{v}\left (g \left ( x \right ) \right )}{\it \_C1} \right ) \right \} \]