23.3.50 problem 52

Internal problem ID [5764]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 52
Date solved : Tuesday, September 30, 2025 at 02:02:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -\left (a^{2}-b \,{\mathrm e}^{x}\right ) y+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 35

ode:=-(a^2-b*exp(x))*y(x)+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 60

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

\begin{align*} y(x)&\to c_1 \operatorname {Gamma}(1-2 a) \operatorname {BesselJ}\left (-2 a,2 \sqrt {b} \sqrt {e^x}\right )+c_2 \operatorname {Gamma}(2 a+1) \operatorname {BesselJ}\left (2 a,2 \sqrt {b} \sqrt {e^x}\right ) \end{align*}

✗ Sympy

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

False