\[ y(x) \left (\frac {\left (\frac {1}{4}-v^2\right ) g'(x)^2}{g(x)}+g'(x)^2+\frac {g^3(x)}{2 g'(x)}-\frac {3 g''(x)^2}{4 g'(x)^2}\right )+y''(x)=0 \] ✗ Mathematica : cpu = 0.375258 (sec), leaf count = 0
DSolve[y[x]*((g^3)[x]/(2*Derivative[1][g][x]) + Derivative[1][g][x]^2 + ((1/4 - v^2)*Derivative[1][g][x]^2)/g[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[y[x]*((g^3)[x]/(2*Derivative[1][g][x]) + Derivative[1][g][x]^2 + ((1/4 - v^2)*Derivative[1][g][x]^2)/g[x] - (3*Derivative[2][g][x]^2)/(4*Derivative[1][g][x]^2)) + Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 0.108 (sec), leaf count = 48
dsolve(diff(diff(y(x),x),x)+(1/2*diff(diff(diff(g(x),x),x),x)/diff(g(x),x)-3/4*diff(diff(g(x),x),x)^2/diff(g(x),x)^2+(1/4-v^2)*diff(g(x),x)^2/g(x)+diff(g(x),x)^2)*y(x)=0,y(x))
\[y \left (x \right ) = \frac {c_{2} \WhittakerW \left (\frac {1}{2} i v^{2}-\frac {1}{8} i, \frac {1}{2}, 2 i g \left (x \right )\right )+c_{1} \WhittakerM \left (\frac {1}{2} i v^{2}-\frac {1}{8} i, \frac {1}{2}, 2 i g \left (x \right )\right )}{\sqrt {\frac {d}{d x}g \left (x \right )}}\]