Internal problem ID [8291]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 711.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {\left (\ln \relax (y) x +\ln \relax (y)-1\right ) y}{x +1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 31
dsolve(diff(y(x),x) = -(ln(y(x))*x+ln(y(x))-1)*y(x)/(x+1),y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{{\mathrm e}^{-x} c_{1}} {\mathrm e}^{-\expIntegral \left (1, -x -1\right ) {\mathrm e}^{-x -1}} \]
✓ Solution by Mathematica
Time used: 0.444 (sec). Leaf size: 24
DSolve[y'[x] == ((1 - Log[y[x]] - x*Log[y[x]])*y[x])/(1 + x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{e^{-x-1} (\text {ExpIntegralEi}(x+1)+e c_1)} \\ \end{align*}