61.34.2 problem 2

Internal problem ID [12687]
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 : Monday, March 31, 2025 at 06:51:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 39

ode:=diff(diff(y(x),x),x)+(a*exp(x)-b)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

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

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq((a*exp(x) - b)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False