59.1.358 problem 365

Internal problem ID [9530]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 365
Date solved : Monday, January 27, 2025 at 06:03:48 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} u^{\prime \prime }+\frac {u}{x^{2}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 29

dsolve(diff(u(x),x$2)+1/x^2*u(x)=0,u(x), singsol=all)
 

\[ u = \sqrt {x}\, \left (c_{1} \sin \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right )+c_{2} \cos \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 42

DSolve[D[u[x],{x,2}]+1/x^2*u[x]==0,u[x],x,IncludeSingularSolutions -> True]
 

\[ u(x)\to \sqrt {x} \left (c_1 \cos \left (\frac {1}{2} \sqrt {3} \log (x)\right )+c_2 \sin \left (\frac {1}{2} \sqrt {3} \log (x)\right )\right ) \]