ODE
\[ x^3 \left (e^{x^2}-k^2\right ) y(x)+x y''(x)-y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.1249 (sec), leaf count = 46
\[\left \{\left \{y(x)\to c_1 \Gamma (1-k) J_{-k}\left (\sqrt {e^{x^2}}\right )+c_2 \Gamma (k+1) J_k\left (\sqrt {e^{x^2}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.076 (sec), leaf count = 25
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\sl J}_{k}\left ({{\rm e}^{{\frac {{x}^{2}}{2}}}}\right )}+{\it \_C2}\,{{\sl Y}_{k}\left ({{\rm e}^{{\frac {{x}^{2}}{2}}}}\right )} \right \} \] Mathematica raw input
DSolve[(E^x^2 - k^2)*x^3*y[x] - y'[x] + x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> BesselJ[-k, Sqrt[E^x^2]]*C[1]*Gamma[1 - k] + BesselJ[k, Sqrt[E^x^2]]*C[2]*Gamma[1 + k]}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)-diff(y(x),x)+x^3*(exp(x^2)-k^2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*BesselJ(k,exp(1/2*x^2))+_C2*BesselY(k,exp(1/2*x^2))