Internal problem ID [8280]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 700.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x)*G(y),0]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {1}{x \left (x y^{2}+1+x \right ) y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 62
dsolve(diff(y(x),x) = 1/x/(x*y(x)^2+1+x)/y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {x \left (2 \LambertW \left (\frac {c_{1} {\mathrm e}^{-\frac {x -1}{2 x}}}{2}\right ) x +x -1\right )}}{x} \\ y \relax (x ) = -\frac {\sqrt {x \left (2 \LambertW \left (\frac {c_{1} {\mathrm e}^{-\frac {x -1}{2 x}}}{2}\right ) x +x -1\right )}}{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.132 (sec). Leaf size: 72
DSolve[y'[x] == 1/(x*y[x]*(1 + x + x*y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {2 x \text {ProductLog}\left (c_1 e^{\frac {1}{2} \left (\frac {1}{x}-1\right )}\right )+x-1}}{\sqrt {x}} \\ y(x)\to \frac {\sqrt {2 x \text {ProductLog}\left (c_1 e^{\frac {1}{2} \left (\frac {1}{x}-1\right )}\right )+x-1}}{\sqrt {x}} \\ \end{align*}