Internal problem ID [8967]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1388.
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 {\left (v \left (v +1\right ) \left (x -1\right )-x \,a^{2}\right ) y}{4 x^{2} \left (x -1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 82
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-1)-a^2*x)/x^2/(x-1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{-\frac {v}{2}} \left (x -1\right )^{-\frac {a}{2}} \hypergeom \left (\left [-\frac {v}{2}-\frac {a}{2}, \frac {1}{2}-\frac {v}{2}-\frac {a}{2}\right ], \left [\frac {1}{2}-v \right ], x\right )+c_{2} x^{\frac {1}{2}+\frac {v}{2}} \left (x -1\right )^{-\frac {a}{2}} \hypergeom \left (\left [1+\frac {v}{2}-\frac {a}{2}, \frac {1}{2}+\frac {v}{2}-\frac {a}{2}\right ], \left [\frac {3}{2}+v \right ], x\right ) \]
✓ Solution by Mathematica
Time used: 0.155 (sec). Leaf size: 109
DSolve[y''[x] == -1/4*((v*(1 + v)*(-1 + x) - a^2*x)*y[x])/((-1 + x)^2*x^2) - ((-1 + 3*x)*y'[x])/(2*(-1 + x)*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(-1)^{-v} (x-1)^{\frac {a+1}{2}} x^{-v/2} \left (c_1 (-1)^v x^{v+\frac {1}{2}} \, _2F_1\left (\frac {1}{2} (a+v+1),\frac {1}{2} (a+v+2);v+\frac {3}{2};x\right )-i c_2 \, _2F_1\left (\frac {a-v}{2},\frac {1}{2} (a-v+1);\frac {1}{2}-v;x\right )\right )}{\sqrt {1-x}} \\ \end{align*}