60.3.264 problem 1269

Internal problem ID [11274]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1269
Date solved : Tuesday, January 28, 2025 at 05:57:53 PM
CAS classification : [_Jacobi]

\begin{align*} 2 x \left (x -1\right ) y^{\prime \prime }+\left (\left (2 v +5\right ) x -2 v -3\right ) y^{\prime }+\left (v +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.350 (sec). Leaf size: 78

dsolve(2*x*(x-1)*diff(diff(y(x),x),x)+((2*v+5)*x-2*v-3)*diff(y(x),x)+(v+1)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {\left (c_{1} \Gamma \left (v +\frac {1}{2}\right )^{2} \left (v +\frac {1}{2}\right ) \operatorname {LegendreP}\left (-\frac {1}{2}, -v -\frac {1}{2}, \frac {-x -1}{x -1}\right )+\sec \left (\pi v \right ) \pi \operatorname {LegendreP}\left (-\frac {1}{2}, v +\frac {1}{2}, \frac {-x -1}{x -1}\right ) c_{2} \right ) x^{-\frac {v}{2}-\frac {1}{4}}}{\sqrt {1-x}\, \Gamma \left (v +\frac {1}{2}\right )} \]

✓ Solution by Mathematica

Time used: 0.126 (sec). Leaf size: 59

DSolve[(1 + v)*y[x] + (-3 - 2*v + (5 + 2*v)*x)*D[y[x],x] + 2*(-1 + x)*x*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},v+1,v+\frac {3}{2},x\right )-i c_2 i^{-2 v} x^{-v-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-v,\frac {1}{2}-v,x\right ) \]