\[ -f(x)+\left (-v^2+x^2+1\right ) y(x)+x^2 y''(x)+3 x y'(x)=0 \] ✓ Mathematica : cpu = 0.0387538 (sec), leaf count = 75
DSolve[-f[x] + (1 - v^2 + x^2)*y[x] + 3*x*Derivative[1][y][x] + x^2*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {\operatorname {BesselJ}(v,x) \int _1^x-\frac {1}{2} \pi \operatorname {BesselY}(v,K[1]) f(K[1])dK[1]+\operatorname {BesselY}(v,x) \int _1^x\frac {1}{2} \pi \operatorname {BesselJ}(v,K[2]) f(K[2])dK[2]}{x}+\frac {c_1 \operatorname {BesselJ}(v,x)}{x}+\frac {c_2 \operatorname {BesselY}(v,x)}{x}\right \}\right \}\] ✓ Maple : cpu = 0.044 (sec), leaf count = 49
dsolve(x^2*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+(-v^2+x^2+1)*y(x)-f(x)=0,y(x))
\[y \left (x \right ) = \frac {\pi \left (\int \operatorname {BesselJ}\left (v , x\right ) f \left (x \right )d x \right ) \operatorname {BesselY}\left (v , x\right )-\pi \left (\int \operatorname {BesselY}\left (v , x\right ) f \left (x \right )d x \right ) \operatorname {BesselJ}\left (v , x\right )+2 \operatorname {BesselY}\left (v , x\right ) c_{1}+2 \operatorname {BesselJ}\left (v , x\right ) c_{2}}{2 x}\]