60.3.239 problem 1244

Internal problem ID [11249]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1244
Date solved : Tuesday, January 28, 2025 at 05:52:46 PM
CAS classification : [_Gegenbauer]

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

Solution by Maple

Time used: 0.120 (sec). Leaf size: 27

dsolve((x^2-1)*diff(diff(y(x),x),x)+2*(n+1)*x*diff(y(x),x)-(v+n+1)*(v-n)*y(x)=0,y(x), singsol=all)
 
\[ y = \left (\operatorname {LegendreP}\left (v , n , x\right ) c_{1} +\operatorname {LegendreQ}\left (v , n , x\right ) c_{2} \right ) \left (x^{2}-1\right )^{-\frac {n}{2}} \]

Solution by Mathematica

Time used: 0.060 (sec). Leaf size: 32

DSolve[(n - v)*(1 + n + v)*y[x] + 2*(1 + n)*x*D[y[x],x] + (-1 + x^2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 
\[ y(x)\to \left (x^2-1\right )^{-n/2} (c_1 P_v^n(x)+c_2 Q_v^n(x)) \]