ODE
\[ \left (e^{2/x}-a^2\right ) y(x)+x^4 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.569911 (sec), leaf count = 100
\[\left \{\left \{y(x)\to \frac {(-1)^{-a} 2^{\frac {3 a}{2}+\frac {1}{2}} \left (-e^{2/x}\right )^{-a/2} \left (e^{2/x}\right )^{a/2} \left ((-1)^a c_1 I_a\left (\sqrt {-e^{2/x}}\right )+c_2 K_a\left (\sqrt {-e^{2/x}}\right )\right )}{\log \left (e^{2/x}\right )}\right \}\right \}\]
Maple ✓
cpu = 0.074 (sec), leaf count = 23
\[ \left \{ y \left ( x \right ) =x \left ( {{\sl Y}_{a}\left ({{\rm e}^{{x}^{-1}}}\right )}{\it \_C2}+{{\sl J}_{a}\left ({{\rm e}^{{x}^{-1}}}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(-a^2 + E^(2/x))*y[x] + x^4*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2^(1/2 + (3*a)/2)*(E^(2/x))^(a/2)*((-1)^a*BesselI[a, Sqrt[-E^(2/x)]]*C[1] + BesselK[a, Sqrt[-E^(2/x)]]*C[2]))/((-1)^a*(-E^(2/x))^(a/2)*Log[E^(2/x)])}}
Maple raw input
dsolve(x^4*diff(diff(y(x),x),x)+(exp(2/x)-a^2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = x*(BesselY(a,exp(1/x))*_C2+BesselJ(a,exp(1/x))*_C1)