23.5.4 problem 3(c)

Internal problem ID [4181]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(c)
Date solved : Monday, January 27, 2025 at 08:41:25 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 32

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

\[ y \left (x \right ) = c_{1} \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 43

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

\[ \{y(x)\}\to c_1 \left (\frac {x^3}{24}-\frac {x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {x^4}{120}-\frac {x^2}{6}+1\right ) \]