23.5.1 problem 2

Internal problem ID [4178]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 2
Date solved : Monday, January 27, 2025 at 08:41:21 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 32

Order:=6; 
dsolve(diff(y(x),x$2)+1/x^2*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \sqrt {x}\, \left (c_{1} x^{-\frac {i \sqrt {3}}{2}}+c_{2} x^{\frac {i \sqrt {3}}{2}}\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 27

AsymptoticDSolveValue[D[y[x],{x,2}]+1/x^2*y[x]==0,{y[x]},{x,0,"6"-1}]
 

\[ \{y(x)\}\to c_1 x^{-(-1)^{2/3}}+c_2 x^{\sqrt [3]{-1}} \]