Internal problem ID [8912]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1333.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {v \left (v +1\right ) y}{4 x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.105 (sec). Leaf size: 45
dsolve(diff(diff(y(x),x),x) = -1/2/x*(3*x-1)/(x-1)*diff(y(x),x)+1/4*v*(v+1)/x^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{-\frac {v}{2}} \hypergeom \left (\left [\frac {1}{2}, -v \right ], \left [\frac {1}{2}-v \right ], x\right )+c_{2} x^{\frac {1}{2}+\frac {v}{2}} \hypergeom \left (\left [\frac {1}{2}, v +1\right ], \left [\frac {3}{2}+v \right ], x\right ) \]
✓ Solution by Mathematica
Time used: 0.059 (sec). Leaf size: 70
DSolve[y''[x] == (v*(1 + v)*y[x])/(4*x^2) - ((-1 + 3*x)*y'[x])/(2*(-1 + x)*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 i^{-v} x^{-v/2} \, _2F_1\left (\frac {1}{2},-v;\frac {1}{2}-v;x\right )+c_2 i^{v+1} x^{\frac {v+1}{2}} \, _2F_1\left (\frac {1}{2},v+1;v+\frac {3}{2};x\right ) \\ \end{align*}