60.3.90 problem 1104

Internal problem ID [11086]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1104
Date solved : Sunday, March 30, 2025 at 07:42:09 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 83

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

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

✓ Mathematica. Time used: 0.07 (sec). Leaf size: 77

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

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

✗ Sympy

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

TypeError : invalid input: 1 - v