Internal problem ID [8906]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1327.
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 {2 y^{\prime }}{x \left (-2+x \right )}+\frac {y}{x^{2} \left (-2+x \right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 85
dsolve(diff(diff(y(x),x),x) = 2/x/(x-2)*diff(y(x),x)-1/x^2/(x-2)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [2-\frac {\sqrt {2}}{2}, 1-\frac {\sqrt {2}}{2}\right ], \left [1-\sqrt {2}\right ], \frac {x}{2}\right ) x^{-\frac {\sqrt {2}}{2}} \left (x -2\right )^{2}+c_{2} \hypergeom \left (\left [2+\frac {\sqrt {2}}{2}, 1+\frac {\sqrt {2}}{2}\right ], \left [1+\sqrt {2}\right ], \frac {x}{2}\right ) x^{\frac {\sqrt {2}}{2}} \left (x -2\right )^{2} \]
✓ Solution by Mathematica
Time used: 0.097 (sec). Leaf size: 105
DSolve[y''[x] == -(y[x]/((-2 + x)*x^2)) + (2*y'[x])/((-2 + x)*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (-\frac {1}{2}\right )^{-\frac {1}{\sqrt {2}}} x^{-\frac {1}{\sqrt {2}}} \left (\left (-\frac {1}{2}\right )^{\sqrt {2}} c_2 x^{\sqrt {2}} \, _2F_1\left (\frac {1}{\sqrt {2}},-1+\frac {1}{\sqrt {2}};1+\sqrt {2};\frac {x}{2}\right )+c_1 \, _2F_1\left (-\frac {1}{\sqrt {2}},-1-\frac {1}{\sqrt {2}};1-\sqrt {2};\frac {x}{2}\right )\right ) \\ \end{align*}