\[ -f(x)+\left (-v^2+x^2+1\right ) y(x)+x^2 y''(x)+3 x y'(x)=0 \] ✓ Mathematica : cpu = 0.0596752 (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]
\[\left \{\left \{y(x)\to \frac {J_v(x) \int _1^x-\frac {1}{2} \pi Y_v(K[1]) f(K[1])dK[1]+Y_v(x) \int _1^x\frac {1}{2} \pi J_v(K[2]) f(K[2])dK[2]}{x}+\frac {c_1 J_v(x)}{x}+\frac {c_2 Y_v(x)}{x}\right \}\right \}\] ✓ Maple : cpu = 0.047 (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))
\[y \left (x \right ) = \frac {-\BesselJ \left (v , x\right ) \pi \left (\int \BesselY \left (v , x\right ) f \left (x \right )d x \right )+\BesselY \left (v , x\right ) \pi \left (\int \BesselJ \left (v , x\right ) f \left (x \right )d x \right )+2 \BesselJ \left (v , x\right ) c_{2}+2 \BesselY \left (v , x\right ) c_{1}}{2 x}\]