ODE
\[ y''(x)-\left (a^2-e^{2 x}\right ) y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0278276 (sec), leaf count = 46
\[\left \{\left \{y(x)\to c_1 \Gamma (1-a) J_{-a}\left (\sqrt {e^{2 x}}\right )+c_2 \Gamma (a+1) J_a\left (\sqrt {e^{2 x}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.061 (sec), leaf count = 17
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\sl J}_{a}\left ({{\rm e}^{x}}\right )}+{\it \_C2}\,{{\sl Y}_{a}\left ({{\rm e}^{x}}\right )} \right \} \] Mathematica raw input
DSolve[-((a^2 - E^(2*x))*y[x]) + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> BesselJ[-a, Sqrt[E^(2*x)]]*C[1]*Gamma[1 - a] + BesselJ[a, Sqrt[E^(2*x)]]*C[2]*Gamma[1 + a]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)-(a^2-exp(2*x))*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*BesselJ(a,exp(x))+_C2*BesselY(a,exp(x))