Internal problem ID [8283]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 703.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y \left (1-x +y x^{2} \ln \relax (x )+y x^{3}-x \ln \relax (x )-x^{2}\right )}{\left (x -1\right ) x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 68
dsolve(diff(y(x),x) = y(x)*(1-x+y(x)*x^2*ln(x)+x^3*y(x)-x*ln(x)-x^2)/(x-1)/x,y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\dilog \relax (x )} {\mathrm e}^{-x}}{x \left (\left (\int -\frac {{\mathrm e}^{\dilog \relax (x )} \left (x +\ln \relax (x )\right ) {\mathrm e}^{-x}}{\left (x -1\right )^{2}}d x \right ) x +c_{1} x -\left (\int -\frac {{\mathrm e}^{\dilog \relax (x )} \left (x +\ln \relax (x )\right ) {\mathrm e}^{-x}}{\left (x -1\right )^{2}}d x \right )-c_{1}\right )} \]
✓ Solution by Mathematica
Time used: 0.875 (sec). Leaf size: 162
DSolve[y'[x] == (y[x]*(1 - x - x^2 - x*Log[x] + x^3*y[x] + x^2*Log[x]*y[x]))/((-1 + x)*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(1-x)^{-\log (x)-1} e^{-\text {PolyLog}(2,x)-x}}{x \left (-\int _1^x-e^{-K[1]-\text {PolyLog}(2,K[1])} (1-K[1])^{-\log (K[1])-2} (K[1]+\log (K[1]))dK[1]+c_1\right )} \\ y(x)\to 0 \\ y(x)\to -\frac {(1-x)^{-\log (x)-1} e^{-\text {PolyLog}(2,x)-x}}{x \int _1^x-e^{-K[1]-\text {PolyLog}(2,K[1])} (1-K[1])^{-\log (K[1])-2} (K[1]+\log (K[1]))dK[1]} \\ \end{align*}