ODE
\[ a y'(x)+y(x) \left (b+c e^x\right )+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0599587 (sec), leaf count = 99
\[\left \{\left \{y(x)\to e^{-\frac {a x}{2}} \left (c_1 \Gamma \left (1-\sqrt {a^2-4 b}\right ) J_{-\sqrt {a^2-4 b}}\left (2 \sqrt {c e^x}\right )+c_2 \Gamma \left (\sqrt {a^2-4 b}+1\right ) J_{\sqrt {a^2-4 b}}\left (2 \sqrt {c e^x}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.068 (sec), leaf count = 53
\[ \left \{ y \left ( x \right ) ={{\rm e}^{-{\frac {ax}{2}}}} \left ( {{\sl Y}_{\sqrt {{a}^{2}-4\,b}}\left (2\,\sqrt {c}{{\rm e}^{x/2}}\right )}{\it \_C2}+{{\sl J}_{\sqrt {{a}^{2}-4\,b}}\left (2\,\sqrt {c}{{\rm e}^{x/2}}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(b + c*E^x)*y[x] + a*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (BesselJ[-Sqrt[a^2 - 4*b], 2*Sqrt[c*E^x]]*C[1]*Gamma[1 - Sqrt[a^2 - 4*b]] + BesselJ[Sqrt[a^2 - 4*b], 2*Sqrt[c*E^x]]*C[2]*Gamma[1 + Sqrt[a^2 - 4*b]])/E^((a*x)/2)}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)+(b+c*exp(x))*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = exp(-1/2*a*x)*(BesselY((a^2-4*b)^(1/2),2*c^(1/2)*exp(1/2*x))*_C2+BesselJ((a^2-4*b)^(1/2),2*c^(1/2)*exp(1/2*x))*_C1)