ODE
\[ -\left (p^2+i x^2\right ) y(x)+x^2 y''(x)+x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0101922 (sec), leaf count = 32
\[\left \{\left \{y(x)\to c_1 J_p\left (-(-1)^{3/4} x\right )+c_2 Y_p\left (-(-1)^{3/4} x\right )\right \}\right \}\]
Maple ✓
cpu = 0.082 (sec), leaf count = 29
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\sl I}_{p}\left ( \left ( -{\frac {1}{2}}-{\frac {i}{2}} \right ) \sqrt {2}x\right )}+{\it \_C2}\,{{\sl K}_{p}\left ( \left ( -{\frac {1}{2}}-{\frac {i}{2}} \right ) \sqrt {2}x\right )} \right \} \] Mathematica raw input
DSolve[-((p^2 + I*x^2)*y[x]) + x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> BesselJ[p, -((-1)^(3/4)*x)]*C[1] + BesselY[p, -((-1)^(3/4)*x)]*C[2]}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)-(p^2+I*x^2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*BesselI(p,(-1/2-1/2*I)*2^(1/2)*x)+_C2*BesselK(p,(-1/2-1/2*I)*2^(1/2)*x)