61.34.1 problem 1

Internal problem ID [12765]
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 : 1
Date solved : Tuesday, January 28, 2025 at 04:18:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.512 (sec). Leaf size: 39

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

\[ y = c_{1} \operatorname {BesselJ}\left (0, \frac {2 \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )+c_{2} \operatorname {BesselY}\left (0, \frac {2 \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right ) \]

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 55

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

\[ y(x)\to c_1 \operatorname {BesselJ}\left (0,\frac {2 \sqrt {a} \sqrt {e^{x \lambda }}}{\lambda }\right )+2 c_2 \operatorname {BesselY}\left (0,\frac {2 \sqrt {a} \sqrt {e^{x \lambda }}}{\lambda }\right ) \]