Internal problem ID [9598]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1269.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
\[ \boxed {2 x \left (x -1\right ) y^{\prime \prime }+\left (\left (2 v +5\right ) x -2 v -3\right ) y^{\prime }+\left (v +1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.25 (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)
\[ y \left (x \right ) = \frac {x^{-\frac {v}{2}-\frac {1}{4}} \left (c_{1} \Gamma \left (v +\frac {1}{2}\right )^{2} \left (v +\frac {1}{2}\right ) \operatorname {LegendreP}\left (-\frac {1}{2}, -v -\frac {1}{2}, \frac {-x -1}{x -1}\right )+\sec \left (\pi v \right ) \pi \operatorname {LegendreP}\left (-\frac {1}{2}, v +\frac {1}{2}, \frac {-x -1}{x -1}\right ) c_{2} \right )}{\sqrt {1-x}\, \Gamma \left (v +\frac {1}{2}\right )} \]
✓ Solution by Mathematica
Time used: 0.121 (sec). Leaf size: 59
DSolve[(1 + v)*y[x] + (-3 - 2*v + (5 + 2*v)*x)*y'[x] + 2*(-1 + x)*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},v+1,v+\frac {3}{2},x\right )-i c_2 i^{-2 v} x^{-v-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-v,\frac {1}{2}-v,x\right ) \]