Internal problem ID [11100]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{x}-b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 39
dsolve(diff(y(x),x$2)+(a*exp(x)-b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (2 \sqrt {b}, 2 \sqrt {a}\, {\mathrm e}^{\frac {x}{2}}\right )+c_{2} \operatorname {BesselY}\left (2 \sqrt {b}, 2 \sqrt {a}\, {\mathrm e}^{\frac {x}{2}}\right ) \]
✓ Solution by Mathematica
Time used: 0.081 (sec). Leaf size: 76
DSolve[y''[x]+(a*Exp[x]-b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {Gamma}\left (1-2 \sqrt {b}\right ) \operatorname {BesselJ}\left (-2 \sqrt {b},2 \sqrt {a} \sqrt {e^x}\right )+c_2 \operatorname {Gamma}\left (2 \sqrt {b}+1\right ) \operatorname {BesselJ}\left (2 \sqrt {b},2 \sqrt {a} \sqrt {e^x}\right ) \]