\[ -f(x)+\left (x^2-v^2\right ) y(x)+x^2 y''(x)+x y'(x)=0 \] ✓ Mathematica : cpu = 0.108728 (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]
\[\left \{\left \{y(x)\to J_v(x) \int _1^x-\frac {\pi Y_v(K[1]) f(K[1])}{2 K[1]}dK[1]+Y_v(x) \int _1^x\frac {\pi J_v(K[2]) f(K[2])}{2 K[2]}dK[2]+c_1 J_v(x)+c_2 Y_v(x)\right \}\right \}\] ✓ Maple : cpu = 0.038 (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))
\[y \left (x \right ) = -\frac {\BesselJ \left (v , x\right ) \pi \left (\int \frac {\BesselY \left (v , x\right ) f \left (x \right )}{x}d x \right )}{2}+\frac {\BesselY \left (v , x\right ) \pi \left (\int \frac {\BesselJ \left (v , x\right ) f \left (x \right )}{x}d x \right )}{2}+\BesselJ \left (v , x\right ) c_{2}+\BesselY \left (v , x\right ) c_{1}\]