\[ -f(x)+\left (x^2-v^2\right ) y(x)+x^2 y''(x)+x y'(x)=0 \] ✓ Mathematica : cpu = 0.067904 (sec), leaf count = 72
DSolve[-f[x] + (-v^2 + x^2)*y[x] + x*Derivative[1][y][x] + x^2*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \operatorname {BesselJ}(v,x) \int _1^x-\frac {\pi \operatorname {BesselY}(v,K[1]) f(K[1])}{2 K[1]}dK[1]+\operatorname {BesselY}(v,x) \int _1^x\frac {\pi \operatorname {BesselJ}(v,K[2]) f(K[2])}{2 K[2]}dK[2]+c_1 \operatorname {BesselJ}(v,x)+c_2 \operatorname {BesselY}(v,x)\right \}\right \}\] ✓ Maple : cpu = 0.036 (sec), leaf count = 49
dsolve(x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(-v^2+x^2)*y(x)-f(x)=0,y(x))
\[y \left (x \right ) = \frac {\pi \left (\int \frac {\operatorname {BesselJ}\left (v , x\right ) f \left (x \right )}{x}d x \right ) \operatorname {BesselY}\left (v , x\right )}{2}-\frac {\pi \left (\int \frac {\operatorname {BesselY}\left (v , x\right ) f \left (x \right )}{x}d x \right ) \operatorname {BesselJ}\left (v , x\right )}{2}+\operatorname {BesselY}\left (v , x\right ) c_{1}+\operatorname {BesselJ}\left (v , x\right ) c_{2}\]