Internal problem ID [8345]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 765.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y \left (-1-\ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right )+\ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right ) x y\right )}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 106
dsolve(diff(y(x),x) = y(x)*(-1-ln((x-1)*(x+1)/x)+ln((x-1)*(x+1)/x)*x*y(x))/x,y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\dilog \left (x +1\right )} x^{\ln \left (x +1\right )} {\mathrm e}^{-\frac {\ln \relax (x )^{2}}{2}} x^{-\ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right )} {\mathrm e}^{-\dilog \relax (x )}}{x \left (\int -\frac {x^{-\ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right )} {\mathrm e}^{-\dilog \relax (x )} {\mathrm e}^{\dilog \left (x +1\right )} x^{\ln \left (x +1\right )} {\mathrm e}^{-\frac {\ln \relax (x )^{2}}{2}} \ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right )}{x}d x +c_{1}\right )} \]
✓ Solution by Mathematica
Time used: 0.885 (sec). Leaf size: 240
DSolve[y'[x] == (y[x]*(-1 - Log[((-1 + x)*(1 + x))/x] + x*Log[((-1 + x)*(1 + x))/x]*y[x]))/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-\frac {\log (x)}{2}+\log (x+1)-\log \left (x-\frac {1}{x}\right )-1} e^{\text {PolyLog}(2,-x)-\text {PolyLog}(2,1-x)}}{-\int _1^xe^{\text {PolyLog}(2,-K[1])-\text {PolyLog}(2,1-K[1])} K[1]^{-\frac {1}{2} \log (K[1])+\log (K[1]+1)-\log \left (K[1]-\frac {1}{K[1]}\right )-1} \log \left (K[1]-\frac {1}{K[1]}\right )dK[1]+c_1} \\ y(x)\to 0 \\ y(x)\to -\frac {x^{-\frac {\log (x)}{2}+\log (x+1)-\log \left (x-\frac {1}{x}\right )-1} e^{\text {PolyLog}(2,-x)-\text {PolyLog}(2,1-x)}}{\int _1^xe^{\text {PolyLog}(2,-K[1])-\text {PolyLog}(2,1-K[1])} K[1]^{-\frac {1}{2} \log (K[1])+\log (K[1]+1)-\log \left (K[1]-\frac {1}{K[1]}\right )-1} \log \left (K[1]-\frac {1}{K[1]}\right )dK[1]} \\ \end{align*}