65.13.3 problem 20.2 (ii)

Internal problem ID [13809]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 20, Series solutions of second order linear equations. Exercises page 195
Problem number : 20.2 (ii)
Date solved : Tuesday, January 28, 2025 at 06:04:32 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 39

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

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 42

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

\[ y(x)\to c_2 \left (\frac {7 x^5}{120}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^4}{8}-\frac {x^2}{2}+1\right ) \]