61.34.5 problem 5

Internal problem ID [12769]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.3-1. Equations with exponential functions
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 08:24:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 1.146 (sec). Leaf size: 73

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

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

✓ Solution by Mathematica

Time used: 0.565 (sec). Leaf size: 152

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

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