23.3.530 problem 536

Internal problem ID [6244]
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 : 536
Date solved : Friday, October 03, 2025 at 01:57:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-a^{2}+{\mathrm e}^{\frac {2}{x}}\right ) y+x^{4} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 23

ode:=(-a^2+exp(2/x))*y(x)+x^4*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (\operatorname {BesselY}\left (a , {\mathrm e}^{\frac {1}{x}}\right ) c_2 +\operatorname {BesselJ}\left (a , {\mathrm e}^{\frac {1}{x}}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.301 (sec). Leaf size: 100

ode=(-a^2 + E^(2/x))*y[x] + x^4*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {(-1)^{-a} 2^{\frac {3 a}{2}+\frac {1}{2}} \left (-e^{2/x}\right )^{-a/2} \left (e^{2/x}\right )^{a/2} \left ((-1)^a c_1 \operatorname {BesselI}\left (a,\sqrt {-e^{2/x}}\right )+c_2 K_a\left (\sqrt {-e^{2/x}}\right )\right )}{\log \left (e^{2/x}\right )} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 2)) + (-a**2 + exp(2/x))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve x**4*Derivative(y(x), (x, 2)) + (-a**2 + exp(2/x))*y(x)