60.3.234 problem 1239

Internal problem ID [11244]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1239
Date solved : Tuesday, January 28, 2025 at 05:42:14 PM
CAS classification : [_Gegenbauer]

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

✓ Solution by Maple

Time used: 0.121 (sec). Leaf size: 35

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

\[ y = c_{1} \operatorname {LegendreP}\left (\frac {\sqrt {1+4 l}}{2}-\frac {1}{2}, x\right )+c_{2} \operatorname {LegendreQ}\left (\frac {\sqrt {1+4 l}}{2}-\frac {1}{2}, x\right ) \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 46

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

\[ y(x)\to c_1 \operatorname {LegendreP}\left (\frac {1}{2} \left (\sqrt {4 l+1}-1\right ),x\right )+c_2 \operatorname {LegendreQ}\left (\frac {1}{2} \left (\sqrt {4 l+1}-1\right ),x\right ) \]