Internal problem ID [9123]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 788.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
\[ \boxed {y^{\prime }+\frac {y \left (\ln \left (x -1\right )+\coth \left (x +1\right ) x -\coth \left (x +1\right ) x^{2} y\right )}{x \ln \left (x -1\right )}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 106
dsolve(diff(y(x),x) = -y(x)*(ln(x-1)+coth(x+1)*x-coth(x+1)*x^2*y(x))/x/ln(x-1),y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\int -\frac {x \cosh \left (x +1\right )+\ln \left (x -1\right ) \sinh \left (x +1\right )}{\ln \left (x -1\right ) x \sinh \left (x +1\right )}d x}}{c_{1} +\int -\frac {x \,{\mathrm e}^{\int -\frac {x \cosh \left (x +1\right )+\ln \left (x -1\right ) \sinh \left (x +1\right )}{\ln \left (x -1\right ) x \sinh \left (x +1\right )}d x} \cosh \left (x +1\right )}{\ln \left (x -1\right ) \sinh \left (x +1\right )}d x} \]
✓ Solution by Mathematica
Time used: 37.644 (sec). Leaf size: 510
DSolve[y'[x] == -((y[x]*(x*Coth[1 + x] + Log[-1 + x] - x^2*Coth[1 + x]*y[x]))/(x*Log[-1 + x])),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*}