60.3.236 problem 1241

Internal problem ID [11246]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1241
Date solved : Tuesday, January 28, 2025 at 05:42:16 PM
CAS classification : [_Gegenbauer]

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

✓ Solution by Maple

Time used: 0.124 (sec). Leaf size: 23

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

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

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 26

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

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