60.3.327 problem 1344

Internal problem ID [11323]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1344
Date solved : Thursday, March 13, 2025 at 08:50:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.228 (sec). Leaf size: 23

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

\[ y = x \left (c_{1} \operatorname {BesselJ}\left (v , {\mathrm e}^{\frac {1}{x}}\right )+c_{2} \operatorname {BesselY}\left (v , {\mathrm e}^{\frac {1}{x}}\right )\right ) \]

✓ Mathematica. Time used: 0.504 (sec). Leaf size: 100

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

\[ y(x)\to \frac {(-1)^{-v} 2^{\frac {3 v}{2}+\frac {1}{2}} \left (-e^{2/x}\right )^{-v/2} \left (e^{2/x}\right )^{v/2} \left (c_1 (-1)^v \operatorname {BesselI}\left (v,\sqrt {-e^{2/x}}\right )+c_2 K_v\left (\sqrt {-e^{2/x}}\right )\right )}{\log \left (e^{2/x}\right )} \]

✗ Sympy

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

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