ODE
\[ y(x) \left (a+b e^x+c e^{2 x}\right )+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.656868 (sec), leaf count = 136
\[\left \{\left \{y(x)\to \left (e^x\right )^{i \sqrt {a}} e^{-i \sqrt {c} e^x} \left (c_1 U\left (\frac {i b}{2 \sqrt {c}}+i \sqrt {a}+\frac {1}{2},2 i \sqrt {a}+1,2 i \sqrt {c} e^x\right )+c_2 L_{-i \sqrt {a}-\frac {i b}{2 \sqrt {c}}-\frac {1}{2}}^{2 i \sqrt {a}}\left (2 i \sqrt {c} e^x\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.375 (sec), leaf count = 58
\[ \left \{ y \left ( x \right ) ={{\rm e}^{-{\frac {x}{2}}}} \left ( {{\sl W}_{{-{\frac {i}{2}}b{\frac {1}{\sqrt {c}}}},\,i\sqrt {a}}\left (2\,i\sqrt {c}{{\rm e}^{x}}\right )}{\it \_C2}+{{\sl M}_{{-{\frac {i}{2}}b{\frac {1}{\sqrt {c}}}},\,i\sqrt {a}}\left (2\,i\sqrt {c}{{\rm e}^{x}}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(a + b*E^x + c*E^(2*x))*y[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> ((E^x)^(I*Sqrt[a])*(C[1]*HypergeometricU[1/2 + I*Sqrt[a] + ((I/2)*b)/Sqrt[c], 1 + (2*I)*Sqrt[a], (2*I)*Sqrt[c]*E^x] + C[2]*LaguerreL[-1/2 - I*Sqrt[a] - ((I/2)*b)/Sqrt[c], (2*I)*Sqrt[a], (2*I)*Sqrt[c]*E^x]))/E^(I*Sqrt[c]*E^x)}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+(a+b*exp(x)+c*exp(2*x))*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = exp(-1/2*x)*(WhittakerW(-1/2*I*b/c^(1/2),I*a^(1/2),2*I*c^(1/2)*exp(x))*_C2+WhittakerM(-1/2*I*b/c^(1/2),I*a^(1/2),2*I*c^(1/2)*exp(x))*_C1)