Internal problem ID [4536]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number: 29.
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 }+n \left (n +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 101
Order:=6; dsolve((1-x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+n*(n+1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {n \left (n +1\right ) x^{2}}{2}+\frac {n \left (n^{3}+2 n^{2}-5 n -6\right ) x^{4}}{24}\right ) y \relax (0)+\left (x -\frac {\left (n^{2}+n -2\right ) x^{3}}{6}+\frac {\left (n^{4}+2 n^{3}-13 n^{2}-14 n +24\right ) x^{5}}{120}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 120
AsymptoticDSolveValue[(1-x^2)*y''[x]-2*x*y'[x]+n*(n+1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {1}{120} \left (n^2+n\right )^2 x^5+\frac {7}{60} \left (-n^2-n\right ) x^5+\frac {1}{6} \left (-n^2-n\right ) x^3+\frac {x^5}{5}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {1}{24} \left (n^2+n\right )^2 x^4+\frac {1}{4} \left (-n^2-n\right ) x^4+\frac {1}{2} \left (-n^2-n\right ) x^2+1\right ) \]