60.2.214 problem 790

Internal problem ID [10801]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 790
Date solved : Monday, January 27, 2025 at 09:52:56 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 {1+x}{x -1}\right )+\coth \left (\frac {1+x}{x -1}\right ) y^{2}-2 \coth \left (\frac {1+x}{x -1}\right ) x^{2} y+\coth \left (\frac {1+x}{x -1}\right ) x^{4}}{\ln \left (\frac {1}{x -1}\right )} \end{align*}

✗ Solution by Maple

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

\[ \text {No solution found} \]

✓ Solution by Mathematica

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

DSolve[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)],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*}