61.34.2 problem 2

Internal problem ID [12766]
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 : 2
Date solved : Tuesday, January 28, 2025 at 04:18:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.388 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

Time used: 0.055 (sec). Leaf size: 76

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

\[ y(x)\to c_1 \operatorname {Gamma}\left (1-2 \sqrt {b}\right ) \operatorname {BesselJ}\left (-2 \sqrt {b},2 \sqrt {a} \sqrt {e^x}\right )+c_2 \operatorname {Gamma}\left (2 \sqrt {b}+1\right ) \operatorname {BesselJ}\left (2 \sqrt {b},2 \sqrt {a} \sqrt {e^x}\right ) \]