60.2.212 problem 788

Internal problem ID [10799]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 788
Date solved : Monday, January 27, 2025 at 09:49:06 PM
CAS classification : [_Bernoulli]

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

Solution by Maple

Time used: 0.010 (sec). Leaf size: 77

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

Solution by Mathematica

Time used: 2.814 (sec). Leaf size: 510

DSolve[D[y[x],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*}