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 : Monday, January 27, 2025 at 01:34:58 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

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

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