49.20.1 problem 4

Internal problem ID [7730]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 182
Problem number : 4
Date solved : Monday, January 27, 2025 at 03:10:50 PM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 33

Order:=8; 
dsolve((1-x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 32

AsymptoticDSolveValue[(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+2*y[x]==0,y[x],{x,0,"8"-1}]
 

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