60.2.206 problem 782

Internal problem ID [10780]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 782
Date solved : Sunday, March 30, 2025 at 06:37:05 PM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 83

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

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

✓ Mathematica. Time used: 0.868 (sec). Leaf size: 104

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

\begin{align*} y(x)\to \frac {\exp \left (\int _1^x\left (-\coth \left (\frac {1}{K[1]}\right ) \log \left (K[1]+\frac {1}{K[1]}\right )-\frac {1}{K[1]}\right )dK[1]\right )}{-\int _1^x\exp \left (\int _1^{K[2]}\left (-\coth \left (\frac {1}{K[1]}\right ) \log \left (K[1]+\frac {1}{K[1]}\right )-\frac {1}{K[1]}\right )dK[1]\right ) \coth \left (\frac {1}{K[2]}\right ) K[2] \log \left (K[2]+\frac {1}{K[2]}\right )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**2 + 1)/x) - x*log((x**2 + 1)/x) - tanh(1/x))*y(x)/(x*tanh(1/x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out