54.2.189 problem 766

Internal problem ID [12063]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 766
Date solved : Wednesday, October 01, 2025 at 12:08:48 AM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 85

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

\[ y = \frac {{\mathrm e}^{-\int \frac {x \ln \left (\frac {x^{2}-1}{x}\right )+\ln \left (x \right )}{x \ln \left (x \right )}d x}}{-\int \frac {{\mathrm e}^{-\int \frac {x \ln \left (\frac {x^{2}-1}{x}\right )+\ln \left (x \right )}{x \ln \left (x \right )}d x} x \ln \left (\frac {x^{2}-1}{x}\right )}{\ln \left (x \right )}d x +c_1} \]

✓ Mathematica. Time used: 0.365 (sec). Leaf size: 110

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

\begin{align*} y(x)&\to \frac {\exp \left (\int _1^x\left (-\frac {\log \left (K[1]-\frac {1}{K[1]}\right )}{\log (K[1])}-\frac {1}{K[1]}\right )dK[1]\right )}{-\int _1^x\frac {\exp \left (\int _1^{K[2]}\left (-\frac {\log \left (K[1]-\frac {1}{K[1]}\right )}{\log (K[1])}-\frac {1}{K[1]}\right )dK[1]\right ) K[2] \log \left (K[2]-\frac {1}{K[2]}\right )}{\log (K[2])}dK[2]+c_1}\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

Timed Out