\[ y(x) \left (\frac {\left (m^2-v^2\right ) g'(x)^2}{g(x)}+g'(x)^2\right )-y'(x) \left (\frac {(2 m-1) g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}\right )+y''(x)=0 \] ✗ Mathematica : cpu = 0.529297 (sec), leaf count = 0
DSolve[y[x]*(Derivative[1][g][x]^2 + ((m^2 - v^2)*Derivative[1][g][x]^2)/g[x]) - Derivative[1][y][x]*(((-1 + 2*m)*Derivative[1][g][x])/g[x] + Derivative[2][g][x]/Derivative[1][g][x]) + Derivative[2][y][x] == 0,y[x],x]
, could not solve
DSolve[y[x]*(Derivative[1][g][x]^2 + ((m^2 - v^2)*Derivative[1][g][x]^2)/g[x]) - Derivative[1][y][x]*(((-1 + 2*m)*Derivative[1][g][x])/g[x] + Derivative[2][g][x]/Derivative[1][g][x]) + Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 0.126 (sec), leaf count = 74
dsolve(diff(diff(y(x),x),x)-(diff(diff(g(x),x),x)/diff(g(x),x)+(2*m-1)*diff(g(x),x)/g(x))*diff(y(x),x)+((m^2-v^2)*diff(g(x),x)^2/g(x)+diff(g(x),x)^2)*y(x)=0,y(x))
\[y \left (x \right ) = g \left (x \right )^{2 m} {\mathrm e}^{-i g \left (x \right )} \left (\KummerM \left (\frac {1}{2} i m^{2}-\frac {1}{2} i v^{2}+m +\frac {1}{2}, 1+2 m , 2 i g \left (x \right )\right ) c_{1}+\KummerU \left (\frac {1}{2} i m^{2}-\frac {1}{2} i v^{2}+m +\frac {1}{2}, 1+2 m , 2 i g \left (x \right )\right ) c_{2}\right )\]