23.3.51 problem 53

Internal problem ID [5765]
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 : 53
Date solved : Tuesday, September 30, 2025 at 02:02:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 46

ode=-((a^2 - E^(2*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-a) \operatorname {BesselJ}\left (-a,\sqrt {e^{2 x}}\right )+c_2 \operatorname {Gamma}(a+1) \operatorname {BesselJ}\left (a,\sqrt {e^{2 x}}\right ) \end{align*}

✗ Sympy

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

False