Internal problem ID [8356]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 776.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y \left (-\ln \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 \ln \left (\frac {1}{x}\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 92
dsolve(diff(y(x),x) = y(x)*(-ln(1/x)-ln((x^2+1)/x)*x+ln((x^2+1)/x)*x^2*y(x))/x/ln(1/x),y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\int -\frac {\ln \left (\frac {x^{2}+1}{x}\right ) x +\ln \left (\frac {1}{x}\right )}{x \ln \left (\frac {1}{x}\right )}d x}}{\int -\frac {{\mathrm e}^{\int -\frac {\ln \left (\frac {x^{2}+1}{x}\right ) x +\ln \left (\frac {1}{x}\right )}{x \ln \left (\frac {1}{x}\right )}d x} x \ln \left (\frac {x^{2}+1}{x}\right )}{\ln \left (\frac {1}{x}\right )}d x +c_{1}} \]
✓ Solution by Mathematica
Time used: 0.65 (sec). Leaf size: 110
DSolve[y'[x] == (y[x]*(-Log[x^(-1)] - x*Log[(1 + x^2)/x] + x^2*Log[(1 + x^2)/x]*y[x]))/(x*Log[x^(-1)]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\exp \left (\int _1^x\left (-\frac {\log \left (K[1]+\frac {1}{K[1]}\right )}{\log \left (\frac {1}{K[1]}\right )}-\frac {1}{K[1]}\right )dK[1]\right )}{-\int _1^x\frac {\exp \left (\int _1^{K[2]}\left (-\frac {\log \left (K[1]+\frac {1}{K[1]}\right )}{\log \left (\frac {1}{K[1]}\right )}-\frac {1}{K[1]}\right )dK[1]\right ) K[2] \log \left (K[2]+\frac {1}{K[2]}\right )}{\log \left (\frac {1}{K[2]}\right )}dK[2]+c_1} \\ y(x)\to 0 \\ \end{align*}