54.3.361 problem 1378

Internal problem ID [12656]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1378
Date solved : Wednesday, October 01, 2025 at 02:19:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 48

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

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

✓ Mathematica. Time used: 0.141 (sec). Leaf size: 90

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

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

✗ Sympy

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

False