64.14.7 problem 7

Internal problem ID [13566]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.1. Exercises page 232
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 05:53:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 44

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

\[ y = \left (1-\frac {1}{6} x^{3}+\frac {3}{40} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {3}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 49

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

\[ y(x)\to c_1 \left (\frac {3 x^5}{40}-\frac {x^3}{6}+1\right )+c_2 \left (\frac {3 x^5}{40}-\frac {x^4}{12}-\frac {x^3}{6}+x\right ) \]