Internal problem ID [9100]
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]
\[ \boxed {y^{\prime }-\frac {y \left (-1-\ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right )+\ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right ) x y\right )}{x}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 48
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)
\[ y \left (x \right ) = \frac {1}{x^{-\ln \left (x +1\right )+\ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right )+1} c_{1} {\mathrm e}^{\frac {\ln \left (x \right )^{2}}{2}} {\mathrm e}^{-\operatorname {dilog}\left (x +1\right )} {\mathrm e}^{\operatorname {dilog}\left (x \right )}+x} \]
✓ Solution by Mathematica
Time used: 0.809 (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 {e^{\operatorname {PolyLog}(2,-x)-\operatorname {PolyLog}(2,1-x)} x^{-\frac {\log (x)}{2}+\log (x+1)-\log \left (x-\frac {1}{x}\right )-1}}{-\int _1^xe^{\operatorname {PolyLog}(2,-K[1])-\operatorname {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 {e^{\operatorname {PolyLog}(2,-x)-\operatorname {PolyLog}(2,1-x)} x^{-\frac {\log (x)}{2}+\log (x+1)-\log \left (x-\frac {1}{x}\right )-1}}{\int _1^xe^{\operatorname {PolyLog}(2,-K[1])-\operatorname {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*}