\[ -y'(x) \left (\frac {(2 v-1) g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}+\frac {2 h'(x)}{h(x)}\right )+y(x) \left (g'(x)^2+\frac {h'(x) \left (\frac {(2 v-1) g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}+\frac {2 h'(x)}{h(x)}\right )}{h(x)}-\frac {h''(x)}{h(x)}\right )+y''(x)=0 \] ✓ Mathematica : cpu = 0.100362 (sec), leaf count = 32
DSolve[-(Derivative[1][y][x]*(((-1 + 2*v)*Derivative[1][g][x])/g[x] + (2*Derivative[1][h][x])/h[x] + Derivative[2][g][x]/Derivative[1][g][x])) + y[x]*(Derivative[1][g][x]^2 + (Derivative[1][h][x]*(((-1 + 2*v)*Derivative[1][g][x])/g[x] + (2*Derivative[1][h][x])/h[x] + Derivative[2][g][x]/Derivative[1][g][x]))/h[x] - Derivative[2][h][x]/h[x]) + Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 h(x) g(x)^v \operatorname {BesselJ}(v,g(x))+c_2 h(x) g(x)^v \operatorname {BesselY}(v,g(x))\right \}\right \}\] ✓ Maple : cpu = 0.09 (sec), leaf count = 24
dsolve(diff(diff(y(x),x),x)-(diff(diff(g(x),x),x)/diff(g(x),x)+(2*v-1)*diff(g(x),x)/g(x)+2*diff(h(x),x)/h(x))*diff(y(x),x)+(diff(h(x),x)/h(x)*(diff(diff(g(x),x),x)/diff(g(x),x)+(2*v-1)*diff(g(x),x)/g(x)+2*diff(h(x),x)/h(x))-diff(diff(h(x),x),x)/h(x)+diff(g(x),x)^2)*y(x)=0,y(x))
\[y \left (x \right ) = h \left (x \right ) g \left (x \right )^{v} \left (\operatorname {BesselY}\left (v , g \left (x \right )\right ) c_{2}+\operatorname {BesselJ}\left (v , g \left (x \right )\right ) c_{1}\right )\]