55.35.1 problem 1

Internal problem ID [14038]
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 : Thursday, October 02, 2025 at 09:09:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 55

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

\begin{align*} 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 ) \end{align*}

✗ Sympy

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

False