34.6.2 problem 2

Internal problem ID [6076]
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 : 2
Date solved : Wednesday, March 05, 2025 at 12:11:13 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} u^{\prime \prime }-\frac {a^{2} u}{x^{{2}/{3}}}&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 41

ode:=diff(diff(u(x),x),x)-a^2/x^(2/3)*u(x) = 0; 
dsolve(ode,u(x), singsol=all);
 

\[ u = \sqrt {x}\, \left (\operatorname {BesselY}\left (\frac {3}{4}, \frac {3 \sqrt {-a^{2}}\, x^{{2}/{3}}}{2}\right ) c_2 +\operatorname {BesselJ}\left (\frac {3}{4}, \frac {3 \sqrt {-a^{2}}\, x^{{2}/{3}}}{2}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.052 (sec). Leaf size: 79

ode=D[u[x],{x,2}]-a^2*x^(-2/3)*u[x]==0; 
ic={}; 
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
 

\[ u(x)\to \frac {3^{3/4} a^{3/4} \sqrt {x} \left (16 c_1 \operatorname {Gamma}\left (\frac {5}{4}\right ) \operatorname {BesselI}\left (-\frac {3}{4},\frac {3}{2} a x^{2/3}\right )+3 (-1)^{3/4} c_2 \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {BesselI}\left (\frac {3}{4},\frac {3}{2} a x^{2/3}\right )\right )}{8 \sqrt {2}} \]

✓ Sympy. Time used: 0.100 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
a = symbols("a") 
u = Function("u") 
ode = Eq(-a**2*u(x)/x**(2/3) + Derivative(u(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=u(x),ics=ics)
 

\[ u{\left (x \right )} = \sqrt {x} \left (C_{1} J_{\frac {3}{4}}\left (\frac {3 x^{\frac {2}{3}} \sqrt {- a^{2}}}{2}\right ) + C_{2} Y_{\frac {3}{4}}\left (\frac {3 x^{\frac {2}{3}} \sqrt {- a^{2}}}{2}\right )\right ) \]