56.4.34 problem 31

Internal problem ID [8923]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 31
Date solved : Monday, January 27, 2025 at 05:20:03 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 32

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 26

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

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