23.3.41 problem 41

Internal problem ID [5755]
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 : 41
Date solved : Tuesday, September 30, 2025 at 02:02:34 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.057 (sec). Leaf size: 59

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

\[ y = \left (\operatorname {BesselJ}\left (\frac {1}{2+k}, \frac {2 \sqrt {a}\, x^{1+\frac {k}{2}}}{2+k}\right ) c_1 +\operatorname {BesselY}\left (\frac {1}{2+k}, \frac {2 \sqrt {a}\, x^{1+\frac {k}{2}}}{2+k}\right ) c_2 \right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 112

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

\begin{align*} y(x)&\to (k+2)^{-\frac {1}{k+2}} \sqrt {x} a^{\frac {1}{2 k+4}} \left (c_1 \operatorname {Gamma}\left (\frac {k+1}{k+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{k+2},\frac {2 \sqrt {a} x^{\frac {k}{2}+1}}{k+2}\right )+c_2 \operatorname {Gamma}\left (1+\frac {1}{k+2}\right ) \operatorname {BesselJ}\left (\frac {1}{k+2},\frac {2 \sqrt {a} x^{\frac {k}{2}+1}}{k+2}\right )\right ) \end{align*}

✗ Sympy

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

TypeError : Symbol object cannot be interpreted as an integer