60.3.306 problem 1323

Internal problem ID [11302]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1323
Date solved : Sunday, March 30, 2025 at 08:11:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }&=-\frac {2 \left (x -2\right ) y^{\prime }}{x \left (x -1\right )}+\frac {2 \left (x +1\right ) y}{x^{2} \left (x -1\right )} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 17

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

\[ y = \frac {c_1 +c_2 \left (x -1\right )^{3}}{x^{2}} \]

✓ Mathematica. Time used: 0.318 (sec). Leaf size: 60

ode=D[y[x],{x,2}] == -2/x*(x-2)/(x-1)*D[y[x],x]+2/x^2*(x+1)/(x-1)*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\left (-c_2 x \left (x^2-3 x+3\right )-3 c_1\right ) \exp \left (-\frac {1}{2} \int _1^x\left (\frac {4}{K[1]}-\frac {2}{K[1]-1}\right )dK[1]\right )}{3 (x-1)} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + (2*x - 4)*Derivative(y(x), x)/(x*(x - 1)) - (2*x + 2)*y(x)/(x**2*(x - 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False