Internal problem ID [8346]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 766.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y \left (-\ln \relax (x )-\ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right ) x +\ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right ) x^{2} y\right )}{x \ln \relax (x )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 85
dsolve(diff(y(x),x) = y(x)*(-ln(x)-x*ln((x-1)*(x+1)/x)+ln((x-1)*(x+1)/x)*x^2*y(x))/x/ln(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\int -\frac {\ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right ) x +\ln \relax (x )}{\ln \relax (x ) x}d x}}{\int -\frac {{\mathrm e}^{\int -\frac {\ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right ) x +\ln \relax (x )}{\ln \relax (x ) x}d x} x \ln \left (\frac {\left (x -1\right ) \left (x +1\right )}{x}\right )}{\ln \relax (x )}d x +c_{1}} \]
✓ Solution by Mathematica
Time used: 0.521 (sec). Leaf size: 210
DSolve[y'[x] == (y[x]*(-Log[x] - x*Log[((-1 + x)*(1 + x))/x] + x^2*Log[((-1 + x)*(1 + x))/x]*y[x]))/(x*Log[x]),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 (K[1])}-\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 (K[1])}-\frac {1}{K[1]}\right )dK[1]\right ) K[2] \log \left (K[2]-\frac {1}{K[2]}\right )}{\log (K[2])}dK[2]+c_1} \\ y(x)\to 0 \\ y(x)\to -\frac {\exp \left (\int _1^x\left (-\frac {\log \left (K[1]-\frac {1}{K[1]}\right )}{\log (K[1])}-\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 (K[1])}-\frac {1}{K[1]}\right )dK[1]\right ) K[2] \log \left (K[2]-\frac {1}{K[2]}\right )}{\log (K[2])}dK[2]} \\ \end{align*}