60.3.321 problem 1327

Internal problem ID [11331]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1327
Date solved : Tuesday, January 28, 2025 at 05:59:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {2 y^{\prime }}{x \left (x -2\right )}-\frac {y}{x^{2} \left (x -2\right )} \end{align*}

✓ Solution by Maple

Time used: 0.350 (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)
 

\[ y = 4 \left (x^{-\frac {\sqrt {2}}{2}}-x^{1-\frac {\sqrt {2}}{2}}+\frac {x^{2-\frac {\sqrt {2}}{2}}}{4}\right ) c_{1} \operatorname {hypergeom}\left (\left [2-\frac {\sqrt {2}}{2}, 1-\frac {\sqrt {2}}{2}\right ], \left [1-\sqrt {2}\right ], \frac {x}{2}\right )+\operatorname {hypergeom}\left (\left [1+\frac {\sqrt {2}}{2}, 2+\frac {\sqrt {2}}{2}\right ], \left [1+\sqrt {2}\right ], \frac {x}{2}\right ) c_{2} \left (x^{2+\frac {\sqrt {2}}{2}}+4 x^{\frac {\sqrt {2}}{2}}-4 x^{1+\frac {\sqrt {2}}{2}}\right ) \]

✓ Solution by Mathematica

Time used: 0.202 (sec). Leaf size: 105

DSolve[D[y[x],{x,2}] == -(y[x]/((-2 + x)*x^2)) + (2*D[y[x],x])/((-2 + x)*x),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (-\frac {1}{2}\right )^{-\frac {1}{\sqrt {2}}} x^{-\frac {1}{\sqrt {2}}} \left (\left (-\frac {1}{2}\right )^{\sqrt {2}} c_2 x^{\sqrt {2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{\sqrt {2}},-1+\frac {1}{\sqrt {2}},1+\sqrt {2},\frac {x}{2}\right )+c_1 \operatorname {Hypergeometric2F1}\left (-\frac {1}{\sqrt {2}},-1-\frac {1}{\sqrt {2}},1-\sqrt {2},\frac {x}{2}\right )\right ) \]