Internal problem ID [5291]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.035 (sec). Leaf size: 49
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);
\[ y \relax (x ) = \left (-\frac {5}{2} \left (x -1\right )-\frac {3}{8} \left (x -1\right )^{2}+\frac {1}{12} \left (x -1\right )^{3}-\frac {5}{192} \left (x -1\right )^{4}+\frac {3}{320} \left (x -1\right )^{5}-\frac {7}{1920} \left (x -1\right )^{6}+\frac {1}{672} \left (x -1\right )^{7}+\mathrm {O}\left (\left (x -1\right )^{8}\right )\right ) c_{2}+\left (1+\left (x -1\right )+\mathrm {O}\left (\left (x -1\right )^{8}\right )\right ) \left (\ln \left (x -1\right ) c_{2}+c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 86
AsymptoticDSolveValue[(1-x^2)*y''[x]-2*x*y'[x]+2*y[x]==0,y[x],{x,1,7}]
\[ y(x)\to c_1 x+c_2 \left (\frac {1}{672} (x-1)^7-\frac {7 (x-1)^6}{1920}+\frac {3}{320} (x-1)^5-\frac {5}{192} (x-1)^4+\frac {1}{12} (x-1)^3-\frac {3}{8} (x-1)^2-2 (x-1)+\frac {1-x}{2}+x \log (x-1)\right ) \]