60.3.240 problem 1245

Internal problem ID [11250]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1245
Date solved : Tuesday, January 28, 2025 at 05:52:48 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: 0.119 (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: 0.045 (sec). Leaf size: 32

DSolve[(-1 + n - v)*(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]
 

\[ y(x)\to \left (x^2-1\right )^{n/2} (c_1 P_v^n(x)+c_2 Q_v^n(x)) \]