Internal problem ID [9717]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1389.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\frac {\left (-1+3 x \right ) y^{\prime }}{2 x \left (x -1\right )}+\frac {\left (-v \left (v +1\right ) \left (x -1\right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}}=0} \]
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 87
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)^2-4*n^2*x)/x^2/(x-1)^2*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (1-x \right )^{n -\frac {1}{2}} \left (x -1\right )^{-n} x^{\frac {1}{4}} \left (\Gamma \left (v +\frac {1}{2}\right )^{2} c_{2} \left (v +\frac {1}{2}\right ) \operatorname {LegendreP}\left (n -\frac {1}{2}, -v -\frac {1}{2}, \frac {-x -1}{x -1}\right )+\sec \left (\pi v \right ) \operatorname {LegendreP}\left (n -\frac {1}{2}, v +\frac {1}{2}, \frac {-x -1}{x -1}\right ) \pi c_{1} \right )}{\Gamma \left (v +\frac {1}{2}\right )} \]
✓ Solution by Mathematica
Time used: 0.4 (sec). Leaf size: 91
DSolve[y''[x] == -1/4*((-(v*(1 + v)*(-1 + x)^2) - 4*n^2*x)*y[x])/((-1 + x)^2*x^2) - ((-1 + 3*x)*y'[x])/(2*(-1 + x)*x),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {(-1)^{-v} (x-1)^{n+\frac {1}{2}} x^{-v/2} \left (c_1 (-1)^v x^{v+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (n+\frac {1}{2},n+v+1,v+\frac {3}{2},x\right )-i c_2 \operatorname {Hypergeometric2F1}\left (n+\frac {1}{2},n-v,\frac {1}{2}-v,x\right )\right )}{\sqrt {1-x}} \]