28.6.15 problem 19

Internal problem ID [7224]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter IX, Special forms of differential equations. Examples XVII. page 247
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 04:25:44 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }+\frac {y}{4 x}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)+1/4*y(x)/x = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 38

ode=D[y[x],{x,2}]+y[x]/(4*x)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \sqrt {x} \left (c_1 \operatorname {BesselJ}\left (1,\sqrt {x}\right )+2 i c_2 \operatorname {BesselY}\left (1,\sqrt {x}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.041 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + y(x)/(4*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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