60.3.20 problem 1020

Internal problem ID [11030]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1020
Date solved : Tuesday, January 28, 2025 at 05:40:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{x}+c \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.566 (sec). Leaf size: 52

dsolve(diff(diff(y(x),x),x)+(a*exp(2*x)+b*exp(x)+c)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {x}{2}} \left (\operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}}, i \sqrt {c}, 2 i \sqrt {a}\, {\mathrm e}^{x}\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}}, i \sqrt {c}, 2 i \sqrt {a}\, {\mathrm e}^{x}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.597 (sec). Leaf size: 144

DSolve[(c + b*E^x + a*E^(2*x))*y[x] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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