49.17.5 problem 1(e)

Internal problem ID [7712]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number : 1(e)
Date solved : Monday, January 27, 2025 at 03:10:23 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*} 1 \end{align*}

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 42

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=1);
 

\[ y = \left (-\frac {5}{2} \left (x -1\right )-\frac {3}{8} \left (x -1\right )^{2}+\frac {1}{12} \left (x -1\right )^{3}-\frac {5}{192} \left (x -1\right )^{4}+\frac {3}{320} \left (x -1\right )^{5}-\frac {7}{1920} \left (x -1\right )^{6}+\frac {1}{672} \left (x -1\right )^{7}+\operatorname {O}\left (\left (x -1\right )^{8}\right )\right ) c_{2} +\left (1+\left (x -1\right )+\operatorname {O}\left (\left (x -1\right )^{8}\right )\right ) \left (c_{2} \ln \left (x -1\right )+c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 86

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

\[ y(x)\to c_1 x+c_2 \left (\frac {1}{672} (x-1)^7-\frac {7 (x-1)^6}{1920}+\frac {3}{320} (x-1)^5-\frac {5}{192} (x-1)^4+\frac {1}{12} (x-1)^3-\frac {3}{8} (x-1)^2-2 (x-1)+\frac {1-x}{2}+x \log (x-1)\right ) \]