60.3.310 problem 1327

Internal problem ID [11306]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1327
Date solved : Sunday, March 30, 2025 at 08:11:09 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*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 122

ode:=diff(diff(y(x),x),x) = 2/x/(x-2)*diff(y(x),x)-1/x^2/(x-2)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = 4 c_1 \left (x^{-\frac {\sqrt {2}}{2}}-x^{1-\frac {\sqrt {2}}{2}}+\frac {x^{2-\frac {\sqrt {2}}{2}}}{4}\right ) \operatorname {hypergeom}\left (\left [2-\frac {\sqrt {2}}{2}, 1-\frac {\sqrt {2}}{2}\right ], \left [1-\sqrt {2}\right ], \frac {x}{2}\right )+\operatorname {hypergeom}\left (\left [2+\frac {\sqrt {2}}{2}, 1+\frac {\sqrt {2}}{2}\right ], \left [1+\sqrt {2}\right ], \frac {x}{2}\right ) c_2 \left (x^{2+\frac {\sqrt {2}}{2}}+4 x^{\frac {\sqrt {2}}{2}}-4 x^{1+\frac {\sqrt {2}}{2}}\right ) \]

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

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - 2*Derivative(y(x), x)/(x*(x - 2)) + y(x)/(x**2*(x - 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False