60.3.242 problem 1247

Internal problem ID [11252]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1247
Date solved : Monday, January 27, 2025 at 11:00:34 PM
CAS classification : [_Gegenbauer]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 27

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

\[ y = c_{1} \left (x -1\right )^{-a +1}+c_{2} \left (x +1\right )^{-a +1} \]

✓ Solution by Mathematica

Time used: 0.863 (sec). Leaf size: 93

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

\[ y(x)\to \left (x^2-1\right )^{-a/2} \exp \left (\int _1^x\frac {K[1]+\sqrt {(a-1)^2}}{K[1]^2-1}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+\sqrt {(a-1)^2}}{K[1]^2-1}dK[1]\right )dK[2]+c_1\right ) \]