\[ -y'(x) \left (\frac {2 f'(x)}{f(x)}-\frac {g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}\right )+y(x) \left (-\frac {f''(x)}{f(x)}+\frac {f'(x) \left (\frac {2 f'(x)}{f(x)}-\frac {g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}\right )}{f(x)}-\frac {v^2 g'(x)^2}{g(x)^2}+g'(x)^2\right )+y''(x)=0 \] ✓ Mathematica : cpu = 0.144912 (sec), leaf count = 36
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]
\[\left \{\left \{y(x)\to c_1 f(x) J_{\sqrt {v^2}}(g(x))+c_2 f(x) Y_{\sqrt {v^2}}(g(x))\right \}\right \}\] ✓ Maple : cpu = 0.09 (sec), leaf count = 20
dsolve(diff(diff(y(x),x),x)-(2*diff(f(x),x)/f(x)+diff(diff(g(x),x),x)/diff(g(x),x)-diff(g(x),x)/g(x))*diff(y(x),x)+(diff(f(x),x)/f(x)*(2*diff(f(x),x)/f(x)+diff(diff(g(x),x),x)/diff(g(x),x)-diff(g(x),x)/g(x))-diff(diff(f(x),x),x)/f(x)-v^2*diff(g(x),x)^2/g(x)^2+diff(g(x),x)^2)*y(x)=0,y(x))
\[y \left (x \right ) = f \left (x \right ) \left (\BesselY \left (v , g \left (x \right )\right ) c_{2}+\BesselJ \left (v , g \left (x \right )\right ) c_{1}\right )\]