60.2.206 problem 782

Internal problem ID [10793]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 782
Date solved : Monday, January 27, 2025 at 09:44:55 PM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 83

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

\[ y = \frac {{\mathrm e}^{-\int \frac {\coth \left (\frac {1}{x}\right ) \ln \left (\frac {x^{2}+1}{x}\right ) x +1}{x}d x}}{-\int \coth \left (\frac {1}{x}\right ) {\mathrm e}^{-\int \frac {\coth \left (\frac {1}{x}\right ) \ln \left (\frac {x^{2}+1}{x}\right ) x +1}{x}d x} \ln \left (\frac {x^{2}+1}{x}\right ) x d x +c_{1}} \]

✓ Solution by Mathematica

Time used: 0.868 (sec). Leaf size: 104

DSolve[D[y[x],x] == (Coth[x^(-1)]*y[x]*(-(x*Log[(1 + x^2)/x]) - Tanh[x^(-1)] + x^2*Log[(1 + x^2)/x]*y[x]))/x,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\exp \left (\int _1^x\left (-\coth \left (\frac {1}{K[1]}\right ) \log \left (K[1]+\frac {1}{K[1]}\right )-\frac {1}{K[1]}\right )dK[1]\right )}{-\int _1^x\exp \left (\int _1^{K[2]}\left (-\coth \left (\frac {1}{K[1]}\right ) \log \left (K[1]+\frac {1}{K[1]}\right )-\frac {1}{K[1]}\right )dK[1]\right ) \coth \left (\frac {1}{K[2]}\right ) K[2] \log \left (K[2]+\frac {1}{K[2]}\right )dK[2]+c_1} \\ y(x)\to 0 \\ \end{align*}