Internal problem ID [8600]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1020.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{x}+c \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.064 (sec). Leaf size: 61
dsolve(diff(diff(y(x),x),x)+(a*exp(2*x)+b*exp(x)+c)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {x}{2}} \WhittakerM \left (-\frac {i b}{2 \sqrt {a}}, i \sqrt {c}, 2 i \sqrt {a}\, {\mathrm e}^{x}\right )+c_{2} {\mathrm e}^{-\frac {x}{2}} \WhittakerW \left (-\frac {i b}{2 \sqrt {a}}, i \sqrt {c}, 2 i \sqrt {a}\, {\mathrm e}^{x}\right ) \]
✓ Solution by Mathematica
Time used: 0.357 (sec). Leaf size: 136
DSolve[(c + b*E^x + a*E^(2*x))*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-i \sqrt {a} e^x} \left (e^x\right )^{i \sqrt {c}} \left (c_1 \text {HypergeometricU}\left (\frac {i b}{2 \sqrt {a}}+i \sqrt {c}+\frac {1}{2},1+2 i \sqrt {c},2 i \sqrt {a} e^x\right )+c_2 \text {LaguerreL}\left (-\frac {i b}{2 \sqrt {a}}-i \sqrt {c}-\frac {1}{2},2 i \sqrt {c},2 i \sqrt {a} e^x\right )\right ) \\ \end{align*}