##### 4.25.28 $$y(x) \left (a+b e^x+c e^{2 x}\right )+y''(x)=0$$

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 = 0.764 (sec), leaf count = 136

$\left \{\left \{y(x)\to \left (e^x\right )^{i \sqrt {a}} e^{-i \sqrt {c} e^x} \left (c_1 U\left (\frac {i b}{2 \sqrt {c}}+i \sqrt {a}+\frac {1}{2},2 i \sqrt {a}+1,2 i \sqrt {c} e^x\right )+c_2 L_{-\frac {i b}{2 \sqrt {c}}-i \sqrt {a}-\frac {1}{2}}^{2 i \sqrt {a}}\left (2 i \sqrt {c} e^x\right )\right )\right \}\right \}$

Maple
cpu = 1.358 (sec), leaf count = 61

$\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-\frac {x}{2}} \WhittakerM \left (-\frac {i b}{2 \sqrt {c}}, i \sqrt {a}, 2 i \sqrt {c}\, {\mathrm e}^{x}\right )+\textit {\_C2} \,{\mathrm e}^{-\frac {x}{2}} \WhittakerW \left (-\frac {i b}{2 \sqrt {c}}, i \sqrt {a}, 2 i \sqrt {c}\, {\mathrm e}^{x}\right )\right ]$ 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)/S
qrt[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))

Maple raw output

[y(x) = _C1*exp(-1/2*x)*WhittakerM(-1/2*I*b/c^(1/2),I*a^(1/2),2*I*c^(1/2)*exp(x)
)+_C2*exp(-1/2*x)*WhittakerW(-1/2*I*b/c^(1/2),I*a^(1/2),2*I*c^(1/2)*exp(x))]