Internal problem ID [8406]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 826.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (3 x y^{2}+x +3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 51
dsolve(diff(y(x),x) = 1/(6*y(x)^2+x)*(3*x*y(x)^2+x+3*y(x)^2)*y(x)/x/(x+1),y(x), singsol=all)
\[ \frac {1}{\frac {1}{y \relax (x )^{2}}+\frac {6}{x}} = \frac {\left ({\mathrm e}^{\RootOf \left (-{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (x +1\right )^{2} \left ({\mathrm e}^{\textit {\_Z}}+9\right )}{2 x}\right )+3 \,{\mathrm e}^{\textit {\_Z}} c_{1}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+9\right )}+9\right ) x}{54} \]
✓ Solution by Mathematica
Time used: 49.131 (sec). Leaf size: 75
DSolve[y'[x] == (y[x]*(x + 3*y[x]^2 + 3*x*y[x]^2))/(x*(1 + x)*(x + 6*y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {x} \sqrt {\text {ProductLog}\left (\frac {6 e^{2 c_1} x}{(x+1)^2}\right )}}{\sqrt {6}} \\ y(x)\to \frac {\sqrt {x} \sqrt {\text {ProductLog}\left (\frac {6 e^{2 c_1} x}{(x+1)^2}\right )}}{\sqrt {6}} \\ y(x)\to 0 \\ \end{align*}