54.2.211 problem 788
Internal
problem
ID
[12085]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
788
Date
solved
:
Wednesday, October 01, 2025 at 12:14:06 AM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime }&=-\frac {y \left (\ln \left (x -1\right )+\coth \left (x +1\right ) x -\coth \left (x +1\right ) x^{2} y\right )}{x \ln \left (x -1\right )} \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 77
ode:=diff(y(x),x) = -y(x)*(ln(x-1)+coth(1+x)*x-coth(1+x)*x^2*y(x))/x/ln(x-1);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {{\mathrm e}^{-\int \frac {\coth \left (x +1\right ) x +\ln \left (x -1\right )}{\ln \left (x -1\right ) x}d x}}{-\int \frac {x \,{\mathrm e}^{-\int \frac {\coth \left (x +1\right ) x +\ln \left (x -1\right )}{\ln \left (x -1\right ) x}d x} \coth \left (x +1\right )}{\ln \left (x -1\right )}d x +c_1}
\]
✓ Mathematica. Time used: 1.827 (sec). Leaf size: 510
ode=D[y[x],x] == -((y[x]*(x*Coth[1 + x] + Log[-1 + x] - x^2*Coth[1 + x]*y[x]))/(x*Log[-1 + x]));
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\exp \left (\int _1^x-\frac {\cosh (K[1]) \left (\left (1+e^2\right ) K[1]+\left (-1+e^2\right ) \log (K[1]-1)\right )+\left (\left (-1+e^2\right ) K[1]+\left (1+e^2\right ) \log (K[1]-1)\right ) \sinh (K[1])}{K[1] \log (K[1]-1) \left (\left (-1+e^2\right ) \cosh (K[1])+\left (1+e^2\right ) \sinh (K[1])\right )}dK[1]\right )}{-\int _1^x\frac {\exp \left (\int _1^{K[2]}-\frac {\cosh (K[1]) \left (\left (1+e^2\right ) K[1]+\left (-1+e^2\right ) \log (K[1]-1)\right )+\left (\left (-1+e^2\right ) K[1]+\left (1+e^2\right ) \log (K[1]-1)\right ) \sinh (K[1])}{K[1] \log (K[1]-1) \left (\left (-1+e^2\right ) \cosh (K[1])+\left (1+e^2\right ) \sinh (K[1])\right )}dK[1]\right ) K[2] \left (\left (1+e^2\right ) \cosh (K[2])+\left (-1+e^2\right ) \sinh (K[2])\right )}{\log (K[2]-1) \left (\left (-1+e^2\right ) \cosh (K[2])+\left (1+e^2\right ) \sinh (K[2])\right )}dK[2]+c_1}\\ y(x)&\to 0\\ y(x)&\to -\frac {\exp \left (\int _1^x-\frac {\cosh (K[1]) \left (\left (1+e^2\right ) K[1]+\left (-1+e^2\right ) \log (K[1]-1)\right )+\left (\left (-1+e^2\right ) K[1]+\left (1+e^2\right ) \log (K[1]-1)\right ) \sinh (K[1])}{K[1] \log (K[1]-1) \left (\left (-1+e^2\right ) \cosh (K[1])+\left (1+e^2\right ) \sinh (K[1])\right )}dK[1]\right )}{\int _1^x\frac {\exp \left (\int _1^{K[2]}-\frac {\cosh (K[1]) \left (\left (1+e^2\right ) K[1]+\left (-1+e^2\right ) \log (K[1]-1)\right )+\left (\left (-1+e^2\right ) K[1]+\left (1+e^2\right ) \log (K[1]-1)\right ) \sinh (K[1])}{K[1] \log (K[1]-1) \left (\left (-1+e^2\right ) \cosh (K[1])+\left (1+e^2\right ) \sinh (K[1])\right )}dK[1]\right ) K[2] \left (\left (1+e^2\right ) \cosh (K[2])+\left (-1+e^2\right ) \sinh (K[2])\right )}{\log (K[2]-1) \left (\left (-1+e^2\right ) \cosh (K[2])+\left (1+e^2\right ) \sinh (K[2])\right )}dK[2]} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) + (-x**2*y(x)/tanh(x + 1) + x/tanh(x + 1) + log(x - 1))*y(x)/(x*log(x - 1)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out