60.3.14 problem 1014

Internal problem ID [11010]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1014
Date solved : Wednesday, March 05, 2025 at 01:37:36 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.705 (sec). Leaf size: 63

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

\[ y = \sqrt {x}\, \left (\operatorname {BesselJ}\left (\frac {1}{a +2}, \frac {2 \sqrt {-c}\, x^{\frac {a}{2}+1}}{a +2}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{a +2}, \frac {2 \sqrt {-c}\, x^{\frac {a}{2}+1}}{a +2}\right ) c_{2} \right ) \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 119

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

\[ y(x)\to (a+2)^{-\frac {1}{a+2}} \sqrt {x} c^{\frac {1}{2 a+4}} \left (c_1 \operatorname {Gamma}\left (\frac {a+1}{a+2}\right ) \operatorname {BesselI}\left (-\frac {1}{a+2},\frac {2 \sqrt {c} x^{\frac {a}{2}+1}}{a+2}\right )+(-1)^{\frac {1}{a+2}} c_2 \operatorname {Gamma}\left (1+\frac {1}{a+2}\right ) \operatorname {BesselI}\left (\frac {1}{a+2},\frac {2 \sqrt {c} x^{\frac {a}{2}+1}}{a+2}\right )\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-c*x**a*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