1.360 problem 365

Internal problem ID [7094]

Book: Collection of Kovacic problems
Section: section 1
Problem number: 365.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_Emden, _Fowler]]

Solve \begin {gather*} \boxed {u^{\prime \prime }+\frac {u}{x^{2}}=0} \end {gather*}

Solution by Maple

Time used: 0.016 (sec). Leaf size: 31

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

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

Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 42

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

\begin{align*} 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 ) \\ \end{align*}