ODE
\[ y''(x)-y(x) \left (a^2-b e^x\right )=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0280964 (sec), leaf count = 60
\[\left \{\left \{y(x)\to c_1 \Gamma (1-2 a) J_{-2 a}\left (2 \sqrt {b} \sqrt {e^x}\right )+c_2 \Gamma (2 a+1) J_{2 a}\left (2 \sqrt {b} \sqrt {e^x}\right )\right \}\right \}\]
Maple ✓
cpu = 0.074 (sec), leaf count = 35
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\sl J}_{2\,a}\left (2\,\sqrt {b}{{\rm e}^{x/2}}\right )}+{\it \_C2}\,{{\sl Y}_{2\,a}\left (2\,\sqrt {b}{{\rm e}^{x/2}}\right )} \right \} \] Mathematica raw input
DSolve[-((a^2 - b*E^x)*y[x]) + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> BesselJ[-2*a, 2*Sqrt[b]*Sqrt[E^x]]*C[1]*Gamma[1 - 2*a] + BesselJ[2*a, 2*Sqrt[b]*Sqrt[E^x]]*C[2]*Gamma[1 + 2*a]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)-(a^2-b*exp(x))*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*BesselJ(2*a,2*b^(1/2)*exp(1/2*x))+_C2*BesselY(2*a,2*b^(1/2)*exp(1/2*x))