54.2.188 problem 765

Internal problem ID [12062]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 765
Date solved : Wednesday, October 01, 2025 at 12:08:37 AM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.065 (sec). Leaf size: 50

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

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

✓ Mathematica. Time used: 0.539 (sec). Leaf size: 240

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

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

✓ Sympy. Time used: 26.944 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = \frac {e^{- \int \frac {\log {\left (x - \frac {1}{x} \right )}}{x}\, dx}}{x \left (C_{1} + e^{- \int \frac {\log {\left (x - \frac {1}{x} \right )}}{x}\, dx}\right )} \]