54.3.89 problem 1103

Internal problem ID [12384]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1103
Date solved : Wednesday, October 01, 2025 at 01:44:11 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }-2 y^{\prime }+a y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 80

ode:=x*diff(diff(y(x),x),x)-2*diff(y(x),x)+a*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (c_1 \left (a x -2\right ) \operatorname {BesselJ}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )+c_2 \left (a x -2\right ) \operatorname {BesselY}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )+2 \sqrt {a}\, \sqrt {x}\, \left (c_1 \operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )+c_2 \operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )\right )\right ) \sqrt {x}}{a} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 56

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

\begin{align*} y(x)&\to 2 a^{3/2} x^{3/2} \left (3 c_1 \operatorname {BesselJ}\left (3,2 \sqrt {a} \sqrt {x}\right )-i c_2 \operatorname {BesselY}\left (3,2 \sqrt {a} \sqrt {x}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = x^{\frac {3}{2}} \left (C_{1} J_{3}\left (2 \sqrt {a} \sqrt {x}\right ) + C_{2} Y_{3}\left (2 \sqrt {a} \sqrt {x}\right )\right ) \]