65.13.1 problem 20.1

Internal problem ID [13807]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 20, Series solutions of second order linear equations. Exercises page 195
Problem number : 20.1
Date solved : Tuesday, January 28, 2025 at 06:04:30 AM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 81

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);
 

\[ y = \left (1-\frac {n \left (n +1\right ) x^{2}}{2}+\frac {n \left (n^{3}+2 n^{2}-5 n -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (n^{2}+n -2\right ) x^{3}}{6}+\frac {\left (n^{4}+2 n^{3}-13 n^{2}-14 n +24\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 120

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

\[ y(x)\to c_2 \left (\frac {1}{120} \left (n^2+n\right )^2 x^5+\frac {7}{60} \left (-n^2-n\right ) x^5+\frac {1}{6} \left (-n^2-n\right ) x^3+\frac {x^5}{5}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {1}{24} \left (n^2+n\right )^2 x^4+\frac {1}{4} \left (-n^2-n\right ) x^4+\frac {1}{2} \left (-n^2-n\right ) x^2+1\right ) \]