60.3.126 problem 1140

Internal problem ID [11122]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1140
Date solved : Sunday, March 30, 2025 at 07:43:17 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 134

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

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

✓ Mathematica. Time used: 0.097 (sec). Leaf size: 120

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

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

✗ Sympy

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

TypeError : invalid input: 1 - b/a