61.34.3 problem 3

Internal problem ID [12767]
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 : 3
Date solved : Tuesday, January 28, 2025 at 08:24:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.362 (sec). Leaf size: 32

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

\[ y = {\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }} \left (c_{1} +\operatorname {Ei}_{1}\left (-\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.888 (sec). Leaf size: 51

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

\[ y(x)\to e^{-\frac {a e^{\lambda x}}{\lambda }} \left (c_2 \int _1^{e^{x \lambda }}\frac {e^{\frac {2 a K[1]}{\lambda }}}{K[1]}dK[1]+c_1\right ) \]