Internal problem ID [9655]
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]]
\[ \boxed {y^{\prime \prime }-\frac {2 y^{\prime }}{x \left (x -2\right )}+\frac {y}{x^{2} \left (x -2\right )}=0} \]
✓ Solution by Maple
Time used: 0.234 (sec). Leaf size: 122
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)
\[ y \left (x \right ) = 4 \left (x^{-\frac {\sqrt {2}}{2}}-x^{1-\frac {\sqrt {2}}{2}}+\frac {x^{2-\frac {\sqrt {2}}{2}}}{4}\right ) c_{1} \operatorname {hypergeom}\left (\left [2-\frac {\sqrt {2}}{2}, 1-\frac {\sqrt {2}}{2}\right ], \left [1-\sqrt {2}\right ], \frac {x}{2}\right )+\operatorname {hypergeom}\left (\left [2+\frac {\sqrt {2}}{2}, 1+\frac {\sqrt {2}}{2}\right ], \left [1+\sqrt {2}\right ], \frac {x}{2}\right ) c_{2} \left (x^{2+\frac {\sqrt {2}}{2}}+4 x^{\frac {\sqrt {2}}{2}}-4 x^{1+\frac {\sqrt {2}}{2}}\right ) \]
✓ Solution by Mathematica
Time used: 0.226 (sec). Leaf size: 105
DSolve[y''[x] == -(y[x]/((-2 + x)*x^2)) + (2*y'[x])/((-2 + x)*x),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (-\frac {1}{2}\right )^{-\frac {1}{\sqrt {2}}} x^{-\frac {1}{\sqrt {2}}} \left (\left (-\frac {1}{2}\right )^{\sqrt {2}} c_2 x^{\sqrt {2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{\sqrt {2}},-1+\frac {1}{\sqrt {2}},1+\sqrt {2},\frac {x}{2}\right )+c_1 \operatorname {Hypergeometric2F1}\left (-\frac {1}{\sqrt {2}},-1-\frac {1}{\sqrt {2}},1-\sqrt {2},\frac {x}{2}\right )\right ) \]