68.2.7 problem Problem 3.7(g)

Internal problem ID [14152]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 3 Bessel functions. Problems page 89
Problem number : Problem 3.7(g)
Date solved : Tuesday, January 28, 2025 at 08:25:28 PM
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*}

✓ Solution by Maple

Time used: 0.211 (sec). Leaf size: 15

dsolve((1-x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+n*(n+1)*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \operatorname {LegendreP}\left (n , x\right )+c_{2} \operatorname {LegendreQ}\left (n , x\right ) \]

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 18

DSolve[(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+n*(n+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \operatorname {LegendreP}(n,x)+c_2 \operatorname {LegendreQ}(n,x) \]