\[ -\frac {f'(x) y'(x)}{f(x)}+y(x) \left (-\frac {f''(x)}{2 f(x)}+\frac {3 f'(x)^2}{4 f(x)^2}+\frac {\left (\frac {1}{4}-v^2\right ) g'(x)^2}{g(x)^2}+g'(x)^2+\frac {g^3(x)}{2 g'(x)}-\frac {3 g''(x)^2}{4 g'(x)^2}\right )+y''(x)=0 \] ✗ Mathematica : cpu = 0.650077 (sec), leaf count = 0
DSolve[-((Derivative[1][f][x]*Derivative[1][y][x])/f[x]) + y[x]*((3*Derivative[1][f][x]^2)/(4*f[x]^2) + (g^3)[x]/(2*Derivative[1][g][x]) + Derivative[1][g][x]^2 + ((1/4 - v^2)*Derivative[1][g][x]^2)/g[x]^2 - Derivative[2][f][x]/(2*f[x]) - (3*Derivative[2][g][x]^2)/(4*Derivative[1][g][x]^2)) + Derivative[2][y][x] == 0,y[x],x]
, could not solve
DSolve[-((Derivative[1][f][x]*Derivative[1][y][x])/f[x]) + y[x]*((3*Derivative[1][f][x]^2)/(4*f[x]^2) + (g^3)[x]/(2*Derivative[1][g][x]) + Derivative[1][g][x]^2 + ((1/4 - v^2)*Derivative[1][g][x]^2)/g[x]^2 - Derivative[2][f][x]/(2*f[x]) - (3*Derivative[2][g][x]^2)/(4*Derivative[1][g][x]^2)) + Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 0.11 (sec), leaf count = 31
dsolve(diff(diff(y(x),x),x)-diff(f(x),x)*diff(y(x),x)/f(x)+(3/4*diff(f(x),x)^2/f(x)^2-1/2*diff(diff(f(x),x),x)/f(x)-3/4*diff(diff(g(x),x),x)^2/diff(g(x),x)^2+1/2*diff(diff(diff(g(x),x),x),x)/diff(g(x),x)+(1/4-v^2)*diff(g(x),x)^2/g(x)^2+diff(g(x),x)^2)*y(x)=0,y(x))
\[y \left (x \right ) = \sqrt {\frac {g \left (x \right ) f \left (x \right )}{\frac {d}{d x}g \left (x \right )}}\, \left (\BesselY \left (v , g \left (x \right )\right ) c_{2}+\BesselJ \left (v , g \left (x \right )\right ) c_{1}\right )\]