60.3.251 problem 1256

Internal problem ID [11261]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1256
Date solved : Tuesday, January 28, 2025 at 05:52:54 PM
CAS classification : [_Jacobi]

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

✓ Solution by Maple

Time used: 0.365 (sec). Leaf size: 51

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

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

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 26

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

\[ y(x)\to c_1 \operatorname {LegendreP}(v,2 x-1)+c_2 \operatorname {LegendreQ}(v,2 x-1) \]