60.2.214 problem 790

Internal problem ID [10788]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 790
Date solved : Sunday, March 30, 2025 at 06:45:10 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=\frac {2 x \ln \left (\frac {1}{x -1}\right )-\coth \left (\frac {x +1}{x -1}\right )+\coth \left (\frac {x +1}{x -1}\right ) y^{2}-2 \coth \left (\frac {x +1}{x -1}\right ) x^{2} y+\coth \left (\frac {x +1}{x -1}\right ) x^{4}}{\ln \left (\frac {1}{x -1}\right )} \end{align*}

✗ Maple

ode:=diff(y(x),x) = (2*x*ln(1/(x-1))-coth((1+x)/(x-1))+coth((1+x)/(x-1))*y(x)^2-2*coth((1+x)/(x-1))*x^2*y(x)+coth((1+x)/(x-1))*x^4)/ln(1/(x-1)); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 9.442 (sec). Leaf size: 228

ode=D[y[x],x] == (-Coth[(1 + x)/(-1 + x)] + x^4*Coth[(1 + x)/(-1 + x)] + 2*x*Log[(-1 + x)^(-1)] - 2*x^2*Coth[(1 + x)/(-1 + x)]*y[x] + Coth[(1 + x)/(-1 + x)]*y[x]^2)/Log[(-1 + x)^(-1)]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {\exp \left (\int _1^x\frac {2 \coth \left (\frac {K[5]+1}{K[5]-1}\right )}{\log \left (\frac {1}{K[5]-1}\right )}dK[5]\right )}{-\int _1^x\frac {\exp \left (\int _1^{K[6]}\frac {2 \coth \left (\frac {K[5]+1}{K[5]-1}\right )}{\log \left (\frac {1}{K[5]-1}\right )}dK[5]\right ) \coth \left (\frac {K[6]+1}{K[6]-1}\right )}{\log \left (\frac {1}{K[6]-1}\right )}dK[6]+c_1}+x^2+1 \\ y(x)\to x^2+1 \\ y(x)\to -\frac {\exp \left (\int _1^x\frac {2 \coth \left (\frac {K[5]+1}{K[5]-1}\right )}{\log \left (\frac {1}{K[5]-1}\right )}dK[5]\right )}{\int _1^x\frac {\exp \left (\int _1^{K[6]}\frac {2 \coth \left (\frac {K[5]+1}{K[5]-1}\right )}{\log \left (\frac {1}{K[5]-1}\right )}dK[5]\right ) \coth \left (\frac {K[6]+1}{K[6]-1}\right )}{\log \left (\frac {1}{K[6]-1}\right )}dK[6]}+x^2+1 \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-x**4/tanh((x + 1)/(x - 1)) + 2*x**2*y(x)/tanh((x + 1)/(x - 1)) - 2*x*log(1/(x - 1)) - y(x)**2/tanh((x + 1)/(x - 1)) + 1/tanh((x + 1)/(x - 1)))/log(1/(x - 1)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

IndexError : Index out of range: a[1]